The Analytical Theory of Heat - Jean Baptiste Joseph Fourier

Mathematical SciencesFrom its pre-historic roots in simple counting to the algorithms powering modern desktop computers, from the genius of Archimedes to the genius of Einstein, advances in mathematical understanding and numerical techniques have been directly responsible for creating the modern world as we know it. This series will provide a library of the most influential publications and writers on mathematics in its broadest sense. As such, it will show not only the deep roots from which modern science and technology have grown, but also the astonishing breadth of application of mathematical techniques in the humanities and social sciences, and in everyday life.

The Analytical Theory of HeatFirst published in 1878, The Analytical Theory of Heat is Alexander Freeman’s English translation of French mathematician Joseph Fourier’s Theorie Analytique de la Chaleur, originally published in French in 1822. In this groundbreaking study, arguing that previous theories of mechanics advanced by such scientific greats as Archimedes, Galileo, Newton and their successors did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to analyse the conduction of heat in solids. Known in scientific circles as the ‘Fourier Series’, this work paved the way for modern mathematical physics. This translation, now reissued, contains footnotes that cross-reference other writings by Fourier and his contemporaries, along with 20 figures and an extensive bibliography. This book will be especially useful for mathematicians who are interested in trigonometric series and their applications.

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The Analytical Theory of Heat

Jean Baptiste Joseph Fourier

CAmBrID GE UNIvErSIt y PrESS

Cambridge New york melbourne madrid Cape town Singapore São Paolo Delhi

Published in the United States of America by Cambridge University Press, New york

www.cambridge.orgInformation on this title: www.cambridge.org/9781108001786

© in this compilation Cambridge University Press 2009

This edition first published 1878This digitally printed version 2009

ISBN 978-1-108-00178-6

This book reproduces the text of the original edition. The content and language reflect the beliefs, practices and terminology of their time, and have not been updated.

THE

ANALYTICAL THEORY OF HEAT

BY-

JOSEPH FOURIER.

TRANSLATED, WITH NOTES,

ALEXANDER FREEMAN, M.A,FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE.

EDITED FOR THE SYNDICS* OF THE UNIVERSITY PRESS.

AT THE UNIVERSITY PRESS.

LONDON: CAMBRIDGE WAREHOUSE, 17, PATERNOSTER ROW.CAMBRIDGE: DEIGHTON, BELL, AND CO.

LEIPZIG: F. A. BROCKHAUS.

1878

[All Rights reserved,]

PREFACE.

IN preparing this version in English of Fourier's

celebrated treatise on Heat, the translator has followed

faithfully the French original. He has, however, ap-

pended brief foot-notes, in which will be found references

to other writings of Fourier and modern authors on

the subject: these are distinguished by the initials A. F.

The notes marked R. L. E. are taken from pencil me-

moranda on the margin of a copy of the work that

formerly belonged to the late Robert Leslie Ellis,

Fellow of Trinity College, and is now in the possession

of St John's College. It was the translator's hope to

have been able to prefix to this treatise a Memoir

of Fourier's life with some account of his writings ;

unforeseen circumstances have however prevented its

completion in time to appear with the present work.

TABLE

CONTENTS OF THE WORK1.

PAflEPRELIMINARY DISCOURSE 1

CHAPTER I.

Introduction.

SECTION I.

STATEMENT OF THE OBJECT OP THE WORK,ART.

I. Object of the theoretical researches 112—10. Different examples, ring, cube, sphere, infinite prism; the variable

temperature at any point whatever is a function of the coordinatesand of the time. The quantity of heat, which during unit of timecrosses a given surface in the interior of the solid, is also a functionof the time elapsed, and of quantities which determine the form andposition of the surface. The object of the theory is to discover thesefunctions 15

II. The three specific elements which must be observed, are the capacity, theconducibility proper or permeability, and the external conducibility orpenetrability. The coefficients which express them may be regarded atfirst as constant numbers, independent of the temperatures . . . 1 0

12. First statement of the problem of the terrestrial temperatures . . 2013—15. "Conditions necessary to applications of the theory. Object of the

experiments 2116—21. The rays of heat which escape from the same point of a surface

have not the same intensity. The intensity of each ray is proportional

1 Each paragraph of the Table indicates the matter treated of in the articlesindicated at the left of that paragraph. The first of these articles begins atthe page marked on the right.

yi TABLE OF CONTENTS.

ART.

to the cosine of the angle which its direction makes with the normal tothe surface. Divers remarks, and considerations on the object and extentof thermological problems, and on the relations of general analysis withthe study of nature

SECTION II.

GENERAL NOTIONS AND PRELIMINARY DEFINITIONS.

22 24. Permanent temperature, thermometer. The temperature denotedby 0 is that of melting ice. The temperature of water boiling in agiven vessel under a given pressure is denoted by 1 . . • • . 2 6

25. The unit which serves to measure quantities of heat, is the heatrequired to liquify a certain mass of ice . . . . . • . 2 7

26. Specific capacity for heat *&•27—29. Temperatures measured by increments of volume or by the addi-

tional quantities of heat. Those cases only are here considered, in whichthe increments of volume are proportional to the increments of thequantity of heat. This condition does not in general exist in liquids ;it is sensibly true for solid bodies whose temperatures differ very muchfrom those which cause the change of state 28

30. Notion of external conducibility ib.31. We may at first regard the quantity of heat lost as proportional to the

temperature. This proposition is not sensibly true except for certainlimits of temperature 29

32—35. The heat lost into the medium consists of several parts. The effectis compound and variable. Luminous heat ib.

36. Measure of the external conducibility . 3 137. Notion of the conducibility proper. This property also may be observed

in liquids . . * ib.38,39. Equilibrium of temperatures. The effect is independent of contact . 3240—49. First notions of radiant heat, and of the equilibrium which is

established in spaces void of air; of the cause of the reflection of raysof heat, or of their retention in bodies; of the mode of communicationbetween the internal molecules; of the law which regulates the inten-sity of the rays emitted. The law is not disturbed by the reflection ofheat . . , . . ib.

50, 51. First notion of the effects of reflected heat . . . . 3 752—56. Eemarks on the statical or dynamical properties of heat. It is the

principle of elasticity. The elastic force of aeriform fluids exactly indi-cates their temperatures . . 39

SECTION III.

PRINCIPLE OF THE COMMUNICATION OF HEAT.

57—59. When two molecules of the same solid are extremely near and atunequal temperatures, the most heated molecule communicates to thatwhich is less heated a quantity of heat exactly expressed by the productof the duration of the instant, of the extremely small difference of thetemperatures, and of a certain function of the distance of the molecules . 41

TABLE OF CONTENTS. Vll

ART. PAGE

60. When a heated body is placed in an aeriform medium at a lower tem-perature, it loses at each instant a quantity of heat which may beregarded in the first researches as proportional to the excess of thetemperature of the surface over the temperature of the medium . . 43

61—64. The propositions enunciated in the two preceding articles are foundedon divers observations. The primary object of the theory is to discoverall the exact consequences of these propositions. We can then measurethe variations of the coefficients, by comparing the results of calculationwith very exact experiments ib.

SECTION IV.

OF THE UNIFORM AND LINEAR MOVEMENT OF HEAT.

65. The permanent temperatures of an infinite solid included between twoparallel planes maintained at fixed temperatures, are expressed by theequation (v - a) e = (b - a) z ; a and b are the temperatures of the twoextreme planes, e their distance, and v the temperature of the section,whose distance from the lower plane is z 45

66, 67. Notion and measure of the flow of heat 4868, 69. Measure of the conducibility proper 5170. Eemarks on the case in which the direct action of the heat extends to

a sensible distance 5371. State of the same solid when the upper plane is exposed to the air . . ib.72. General conditions of the linear movement of heat 55

SECTION V.

LAW OF THE PERMANENT TEMPERATURES IN A PRISM OF SMALL THICKNESS.

73—80. Equation of the linear movement of heat in the prism. Differentconsequences of this equation 56

SECTION VI.

THE HEATING OF CLOSED SPACES.

81—84. The final state of the solid boundary which encloses the spaceheated by a surface by maintained at the temperature a, is expressed bythe following equation :

p{)

The value of P is - ( % + ^ + -^ J , m is the temperature of the internals \h K H)air, n the temperature of the external air, g, h, H measure respectivelythe penetrability of the heated surface <r, that of the inner surface of theboundary s, and that of the external surface s; e is the thickness of theboundary, and K its conducibility proper . . . . . . . 62

85, 86. Remarkable consequences of the preceding equation . . . . 65

87—91. Measure of the quantity of heat requisite to retain at a constanttemperature, a body whose surface is protected from the external air by

viii TABLE OF CONTENTS.

ART. PAGE

several successive envelopes. Bemarkable effects of the separation of thesurfaces. These results applicable to many different problems . . 67

SECTION VII.

OF THE UNIFORM MOVEMENT OF HEAT IN THREE DIMENSIONS.

92, 93. The permanent temperatures of a solid enclosed between six rec-tangular planes are expressed by the equation

v — A + ax + by + cz.

x, y, z are the coordinates of any point, whose temperature is v; A, a,&, c are constant numbers. If the extreme planes are maintained by anycauses at fixed temperatures which satisfy the preceding equation, thefinal system of all the internal temperatures will be expressed by thesame equation • . . . . 7 3

94, 95. Measure of the flow of heat in this prism 75

SECTION VIII.

MEASURE OF THE MOVEMENT OF HEAT AT A GIVEN POINT OF A GIVEN SOLID.

96—99. The variable system of temperatures of a solid is supposed to beexpressed by the equation v = F (x, y, z, t), where v denotes the variabletemperature which would be observed after the time t had elapsed, at thepoint whose coordinates are x, y, z. Formation of the analytical expres-sion of the flow of heat in a given direction within the solid . . . 78

100. Application of the preceding theorem to the case in which the functionF is e~st cos x cos y cos z • . . . . 82

CHAPTER II.

Equation of the Movement of Heat.

SECTION I.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A RING.

101—105. The variable movement of heat in a ring is expressed by theequation

cWCDdx' CDSV'

The arc x measures the distance of a section from the origin 0; v isthe temperature which that section acquires after the lapse of the time t;K, C\ D, h are the specific coefficients; S is the area of the section, bythe revolution of which the ring is generated; I is the perimeter ofthe section . . . DK

TABLE OF CONTENTS. ix

ART. PAGE106—110. The temperatures at points situated at equal distances are

represented by the terms of a recurring series. Observation of thetemperatures vlt v2> v3 of three consecutive points gives the measure

of the ratio £ : we have t ^ 8 = 3, «*-g« + l = 0. and£=fK V K l£: we have = 3, « g « + l = 0. and£=f f ^ Y -K V2 K l\\logej

The distance between two consecutive points is X, and log w is the decimallogarithm of one of the two values of w . : . , . . 8 6

SECTION II.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID SPHERE.

Ill—113. x denoting the radius of any shell, the movement of heat in thesphere is expressed by the equation

2 dv\* dx)dt CD W ' '

114—117. Conditions relative to the state of the surface and to the initialstate of the solid 92

SECTION III.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID CYLINDER.

118—120. The temperatures of the solid are determined by three equations;the first relates to the internal temperatures, the second expresses thecontinuous state of the surface, the third expresses the initial state ofthe solid 95

SECTION IV.

EQUATIONS OF THE VARIED MOVEMENT OF HEAT IN A SOLID PRISMOF INFINITE LENGTH.

121—123. The system of fixed temperatures satisfies the equation

d2v 6N_ d^v_0

v is the temperature at a point whose coordinates are x , y , z . . . 9 7124, 125. Equation relative to the state of the surface and to that of the

first section 99

SECTION V.

EQUATIONS OP THE VARIED MOVEMENT OF HEAT IN A SOLID CUBE.

126—131. The system of variable temperatures is determined by three

equations; one expresses the internal state, the second relates to the

state of the surface, and the third expresses the initial state . . . 1 0 1

F. H. ^

TABLE OF CONTENTS.

SECTION VI.

GENERAL EQUATION OF THE PROPAGATION OF HEAT IN THE INTERIOR

OF SOLIDS,

ART.

132—139. Elementary proof of properties of the uniform movement of heatin a solid enclosed between six orthogonal planes, the constant tem-peratures being expressed by the linear equation,

v=A - ax-by -cz.The temperatures cannot change, since each point of the solid receivesas much heat as it gives off. The quantity of heat which during theunit of time crosses a plane at right angles to the axis of z is the same,through whatever point of that axis the plane passes. The value of thiscommon flow is that which would exist, if the coefficients a and bwere nul 104

140,141. Analytical expression of the flow in the interior of any solid. Theciv

equation of the temperatures being v=f(x, y, z, t) the function -Kw-j-

expresses the quantity of heat which during the instant dt crosses aninfinitely small area w perpendicular to the axis of z, at the point whosecoordinates are x, y, z, and whose temperature is v after the time thas elapsed 109

142—145. It is easy to derive from the foregoing theorem the generalequation of the movement of heat, namely

dv_K fd*v . d2v . d*v\dt~'CJ

SECTION VII.

GENERAL EQUATION BELATIVE TO THE SURFACE.

146—154. It is proved that the variable temperatures at points on thesurface of a body, which is cooling in air, satisfy the equation

dv dv dv hmdx+ndy'J~Pdz + KVq = 0' mdx + ndy+Pdz=z0>

being the differential equation of the surface which bounds the solid,and q being equal to (m* + n"+2>9)?. To discover this equation weconsider a molecule of the envelop which bounds the solid, and we expressthe fact that the temperature of this element does not change by a finitemagnitude during an infinitely small instant. This condition holds andcontinues to exist after that the regular action of the medium has beenexerted during a very small instant. Any form may be given to theelement of the envelop. The case in which the molecule is formed byrectangular sections presents remarkable properties. In the most simplecase, which is that in which the base is parallel to the tangent plane,the truth of the equation is evident .

• • • •

TABLE OF CONTENTS. xi

SECTION VIII.

APPLICATION OF THE GENERAL EQUATIONS.ABT. PAGE

155, 156. In applying the general equation (A) to the case of the cylinderand of the sphere, we find the same equations as those of Section III.and of Section II. of this chapter 123

SECTION IX.

GENERAL EEMARKS.

157—162. Fundamental considerations on the nature of the quantitiesx, t, v, K, h, C, D, which enter into all the analytical expressions of theTheory of Heat. Each of these quantities has an exponent of dimensionwhich relates to the length, or to the duration, or to the temperature.These exponents are found by making the units of measure vary , . 126

CHAPTER III.

Propagation of Heat in an infinite rectangular solid.

SECTION I.

STATEMENT OF THE PROBLEM.

163—166. The constant temperatures of a rectangular plate included be-tween two parallel infinite sides, maintained at the temperature 0, are

expressed by the equation ^~2 + ^ ~ 2 ~ 0 131

167—170. If we consider the state of the plate at a very great distance fromthe transverse edge, the ratio of the temperatures of two points whosecoordinates are Xj, y and x2, y changes according as the value of yincreases; x1 and x.2 preserving their respective values. The ratio hasa limit to which it approaches more and more, and when y is infinite,it is expressed by the product of a function of x and of a function of y.This remark suffices to disclose the general form of v, namely,

v = 3?~^aie~®l~1)x. cos (2i- l).y.

It is easy to ascertain how the movement of heat in the plate iseffected . - 134

TABLE OF CONTENTS.

SECTION II.

FIRST EXAMPLE OF THE USE OF TRIGONOMETRIC SERIES IN THE

THEORY OF HEAT.

ART. .171__178. Investigation of the coefficients in the equation

l = a cos cc + 6 cos Bx + c cos 5x+d cos 7x + etc.

From which we conclude

* 1 ^

or etc 137

SECTION III.

REMARKS ON THESE SERIES.

179 181. To find the value of the series which forms the second member,the number m of terms is supposed to be limited, and the series becomesa function of x and m. This function is developed according to powers ofthe reciprocal of m, and m is made infinite 145

182—184. The same process is applied to several other series . . . 147185—188. In the preceding development, which gives the value of the

function of x and m, we determine rigorously the limits within which thesum of all the terms is included, starting from a given term . . . 1 5 0

189. Very simple process for forming the series

SECTION IV.

GENERAL SOLUTION.

190, 191. Analytical expression of the movement of heat in a rectangularslab; it is decomposed into simple movements 154

192—195. Measure of the quantity of heat which crosses an edge or sideparallel or perpendicular to the base. This expression of the flow sufficesto verify the solution , -y§§

196-199. Consequences of this solution. The rectangular slab must beconsidered as forming part of an infinite plane; the solution expressesthe permanent temperatures at all points of this plane . . . . 1 5 9

200—204. It is proved that the problem proposed admits of no other solu-tion different from that which we have just stated 161

TABLE OF CONTENTS. Xlll

SECTION V.

FINITE EXPRESSION OF THE KESULT OF THE SOLUTION.ART. PAGE

205, 206. The temperature at a point of the rectangular slab whose co-ordinates are x and y, is expressed thus

2 D \ex-e~xJ

SECTION VI.

DEVELOPMENT OF AN ARBITRARY FUNCTION IN TRIGONOMETRIC SERIES.

207—214. The development obtained by determining the values of the un-known coefficients in the following equations infinite in number:

2 >TT

The value of the general coefficient a< is — / dx<f> (x) sin ix. Whence weIT J Q

derive the very simple theorem

166

To solve these equations, we first suppose the number of equations to bem, and that the number of unknowns a, bt c, d, &c. is m only, omittingall the subsequent terms. The unknowns are determined for a certainvalue of the number m, and the limits to which the values of the coeffi-cients continually approach are sought; these limits are the quantitieswhich it is required to determine. Expression of the values of a, 6, c, d,&c. when m is infinite . . . . . . . . . * 168

215, 216. The function <p(x) developed under the form

a sin x + b sin 2x + c sin Sx + d sin 4# + &c,

which is first supposed to contain only odd powers of x . . . . 179217, 218. Different expression of the same development. Application to the

function ex-e~x 181219—221. Any function whatever 0 (x) may be developed under the form

—0(#) = sinsc P da<p{a)8ma + sin2xj da <p(a) sin2a + sin Sx C da<p(a) sin3a + &c,

whence ^ <p(x) = '2 s i n i x f da (p(a) s i n i a . . . . 1 8 4

222, 223. Application of the theorem : from it is derived the remarkableseries,

^ cos * = =--5 sin a;+ a-= sin 45? + ^-= sin 7*+ =-5 sin 9* +<&c. . . 1884 l . O O.O O . I 4.u

xiv TABLE OF CONTENTS.

AKT.

224, 225. Second theorem on the development of functions in trigono-metrical series:

^i//(#)=2 cosix ["dacos ia\p(a).* i=0 JO

Applications : from it we derive the remarkable series

1 . 1 cos2se cos4# cos 6x D _ r t

i 7 r s i n a ! = _ _ _ _ _ _ _ _ _ _ _ _ _ & « . . . . 190

226—230. The preceding theorems are applicable to discontinuous functions,and solve the problems which are based upon the analysis of DanielBernoulli in the problem of vibrating cords. The value of the series,

sin x versin a + ^ sin 2x versin 2a + ~ sin 3x versin 3a+&c.,

is z, if we attribute to # a quantity greater than 0 and less than a; and

the value of the series is 0, if x is any quantity included between a and \ir.Application to other remarkable examples; curved lines or surfaces whichcoincide in a part of their course, and differ in all the other parts . . 193

231—233. Any function whatever, F{x), may be developed in the form

[x)~ + | ^ s i n 5 C + &2 s i n 2% + b3 sin

Each of the coefficients is a definite integral. We have in general

2vA = / dxF(x)t Trcii = / UdxF(x) cos ixt

and irt>i— \ dxF(x) sin ix.J—n

We thus form the general theorem, which is one of the chief elements ofour analysis:

'(x) = 2 (cos ix / daF(a) cos ia + sin ix J daF(a) sin ia ) ,

i=+co /» + w

or 2irF(x) = S / ( i a ^ a ) c o s ( ^ - i a ) 199

234. The values of F(x) which correspond to values of x includedbetween - it and -f IT must be regarded as entirely arbitrary. We mayalso choose any limits whatever for x . . . . . . . 204

235. Divers remarks on the use of developments in trigonometric series . 206

SECTION VII.

APPLICATION TO THE ACTUAL PROBLEM.

236. 237. Expression of the permanent temperature in the infinite rectangularslab, the state of the transverse edge being represented by an arbitraryfunction . . . . . . . ' ^—

TABLE OF CONTENTS. XV

CHAPTER IV.

Of the linear and varied Movement of Heat in a ring.

SECTION I.

GENERAL SOLUTION OF THE PROBLEM.ART. PAGE

238—241. The variable movement which we are considering is composed ofsimple movements. In each of these movements, the temperatures pre-serve their primitive ratios, and decrease with the time, as the ordinates vof a line whose equation is v = A.e~mt, Formation of the general ex-pression 213

242—244. Application to some remarkable examples. Different consequencesof the solution 218

245, 246. The system of temperatures converges rapidly towards a regularand final state, expressed by the first part of the integral. The sum ofthe temperatures of two points diametrically opposed is then the same,whatever be the position of the diameter. It is equal to the mean tem-perature. In each simple movement, the circumference is divided byequidistant nodes. All these partial movements successively disappear,except the first; and in general the heat distributed throughout the solidassumes a regular disposition, independent of the initial state . . 221

SECTION II.

OP THE COMMUNICATION OF HEAT BETWEEN SEPARATE MASSES.

247—250. Of the communication of heat between two masses. Expressionof the variable temperatures. Remark on the value of the coefficientwhich measures the conducibility 225

251—255. Of the communication of heat between n separate masses, ar-ranged in a straight line. Expression of the variable temperature of eachmass; it is given as a function of the time elapsed, of the coefficientwhich measures the conducibility, and of all the initial temperaturesregarded as arbitrary . 228

256, 257. Remarkable consequences of this solution 236258. Application to the case in which the number of masses is infinite. . 237259—266. Of thB communication of heat between n separate masses arranged

circularly. Differential equations suitable to the problem; integration ofthese equations. The variable temperature of each of the masses is ex-pressed as a function of the coefficient which measures the conducibility,of the time which has elapsed since the instant when the communicationbegan, and of all the initial temperatures, which are arbitrary; but inorder to determine these functions completely, it is necessary to effectthe elimination of the coefficients 238

267—271. Elimination of the coefficients in the equations which containthese unknown quantities and the given initial temperatures . . . 247

x v i TABLE OF CONTENTS.

ART. P A G E

272, 273. Formation of the general solut ion: analytical expression of the

' resu l t _ 2 5 3

274—276. Application and consequences of this solution . . . . 255277, 278. Examination of the case in which the number n is supposed infinite.

We obtain the solution relative to a solid ring, set forth in Article 241,and the theorem of Article 234. We thus ascertain the origin of theanalysis which we have employed to solve the equation relating to con-tinuous bodies 2^9

279. Analytical expression of the two preceding results . . . . 262280 282. I t is proved that the problem of the movement of heat in a ring

dv , d2va d m i t s n o o t h e r s o l u t i o n . T h e i n t e g r a l o f t h e e q u a t i o n - ^ = k ^ i s

e v i d e n t l y t h e m o s t g e n e r a l w h i c h c a n b e f o r m e d . . . . . 2 6 3

CHAPTER V.

Of the Propagation of Heat in a solid sphere.

SECTION I.

GENERAL SOLUTION.

283—289. The ratio of the variable temperatures of two points in the solidis in the first place considered to approach continually a definite limit.

This remark leads to the equation v = A e^KnH, which expressesx

the simple movement of heat in the sphere. The number n has aninfinity of values given by the definite equation — — 1- hX. The

tan 11 Ji.radius of the sphere is denoted by X, and the radius of any concentricsphere, whose temperature is v after the lapse of the time t, by x; hand K are the specific coefficients; A is any constant. Constructionsadapted to disclose the nature of the definite equation, the limits andvalues of its roots 268

290—292. Formation of the general solution; final state of the solid . . 274293. Application to the case in which the sphere has been heated by a pro-

longed immersion 277

SECTION II.

DlFFEKENT REMARKS ON THIS SOLUTION.

294—296. Kesults relative to spheres of small radius, and to the final tern-peratures of any sphere • • • . . . . , 279

298—300. Variable temperature of a thermometer plunged into a liquidwhich is cooling freely. Application of the results to the comparison anduse of thermometers . . . . . .

TABLE OF CONTENTS. xvii

AET. PAGE

301. Expression of the mean temperature of the sphere as a function of thetime elapsed 286

302—304. Application to spheres of very great radius, and to those in whichthe radius is very small 287

305. Remark on the nature of the definite equation which gives all the valuesof n . 289

CHAPTER VI.

Of the Movement of Heat in a solid cylinder.

306, 307. We remark in the first place that the ratio of the variable tem-peratures of two points of the solid approaches continually a definitelimit, and by this we ascertain the expression of the simple movement.The function of x which is one of the factors of this expression is givenby a differential equation of the second order. A number g enters intothis function, and must satisfy a definite equation 291

308, 309. Analysis of this equation. By means of the principal theorems ofalgebra, it is proved that all the roots of the equation are real . . . 294

310. The function u of the variable x is expressed by

u =— / dr cos (xjg sin r);

and the definite equation is hu + — =0, giving to a; its complete value X 296

311, 312. The development of the function <f>{z) being represented by

the value of the series

22.42.6s Ci>

is — f du$(t sinw).TTJQ

Remark on this use of definite integrals 298313. Expression of the function u of the variable a as a continued fraction . 300314. Formation of the general solution 301315—318. Statement of the analysis which determines the values of the co-

efficients , 3 0 3

319. General solution 3 0 8

320. Consequences of the solution 3°9

xviii TABLE OF CONTENTS.

CHAPTER VII.

Propagation of Heat in a rectangular prism.AET.

321—323. Expression of the simple movement determined by the general

properties of heat, and by the form of the solid. Jnto Jhis expression

enters an arc e which satisfies a transcendental equation, all of whose

roots are real 311324. All the unknown coefficients are determined by definite integrals . 313

325. General solution of the problem 314326. 327. The problem proposed admits no other solution . . . . 315

328, 329. Temperatures at points on the axis of the prism . . . . 317330. Application to the case in which the thickness of the prism is very

small 318331. The solution shews how the uniform movement of heat is established

in the interior of the solid 319332. Application to prisms, the dimensions of whose bases are large . . 322

CHAPTER VIII.

Of the Movement of Heat in a solid cube.

333, 334. Expression of the simple movement. Into it enters an arc ewhich must satisfy a trigonometric equation all of whose roots are real . 323

335, 336. Formation of the general solution 324337. The problem can admit no other solution 327338. Consequence of the solution {ft.339. Expression of the mean temperature 328340. Comparison of the final movement of heat in the cube, with the

movement which takes place in the sphere 329341. Application to the simple case considered in Art. 100 . . . . 331

CHAPTER IX.

Of the Diffusion of Heat.

SECTION I.

OF THE FKEE MOVEMENT OF HEAT IN AN INFINITE LINE.

342—347. We consider the linear movement of heat in an infinite line, apart of which has been heated; the initial state is represented byv = F(x). The following theorem is proved :

/•« TooF(x)= J dq cos qx I daF (a) cos get.

Jo •'o

TABLE OF CONTENTS. XIX

ART, PAGE

The function F (x) satisfies the condition F (x) = F( - x). Expression ofthe variable temperatures 333

348. Application to the case in which all the points of the part heatedhave received the same initial temperature. The integral

"dq .— sin q cos qx is JTT,

if we give to x a value included between 1 and - 1.The definite integral has a nul value, if x is not included between

1 and - 1 338349. Application to the case in which the heating given results from the

final state which the action of a source of heat determines . . . 339350. Discontinuous values of the function expressed by the integral

8 4 °351—353. We consider the linear movement of heat in a line whose initial

temperatures are represented by v=f(x) at the distance x to the rightof the origin, and by v=-f(x) at the distance x to the left of the origin.Expression of the variable temperature at any point. The solutionderived from the analysis which expresses the movement of heat in aninfinite line ib.

354. Expression of the variable temperatures when the initial state of thepart heated is expressed by an entirely arbitrary function . . . 343

355—358. The developments of functions in sines or cosines of multiple arcsare transformed into definite integrals 345

359. The following theorem is proved :

-f(x)= I dq sin qx I daf (a) sin qa.* Jo Jo

The function / (x) satisfies the condition:

f(-x) = -f(x) 348

360—362. Use of the preceding results. Proof of the theorem expressedby the general equation:

ir(p (x) = I da <f> (a) I dq cos (qx - qa).7-oo Jo

This equation is evidently included in equation (II) stated in Art. 234.(See Art. 397) t&.

363. The foregoing solution shews also the variable movement of heat in aninfinite line, one point of which is submitted to a constant temperature . 352

364. The same problem nlay also be solved by means of 'another form of theintegral. Formation of this integral 354

365. 366. Application of the solution to an infinite prism, whose initialtemperatures are nul. Eemarkable consequences 356

367—369. The same integral applies to the problem of the diffusion of heat.The solution which we derive from it agrees with that which has beenstated in Articles 347,348 . . . . . . . . 362

x x TABLE OF CONTENTS.

ART. . P A G E

370, 371. Remarks on different forms of the integral of the equation

dw^&u 365

SECTION II.

OF THE FREE MOVEMENT OF HEAT IN AN INFINITE SOLID.

372—376. The expression for the variable movement of heat in an infinitesolid mass, according to three dimensions, is derived immediately fromthat of the linear movement. The integral of the equation

dv _ d2v d2v d?v~dt = dx2 + dy2 + d?

solves the proposed problem. It cannot have a more extended integral;it is derived also from the particular value

v = e~n2t cos nx,or from this:

2

which both satisfy the equation — = -—-2 . The generality of the in-tegrals obtained is founded upon the following proposition, which may beregarded as self-evident. Two functions of the variables x, y, z, t arenecessarily identical, if they satisfy the differential equation

dv _ d2v d2v dHdt dx2 dy2 dz2 *

and if at the same time they have the same value for a certain valueof t 368

377—382. The heat contained in a part of an infinite prism, all the otherpoints of which have nul initial temperature, begins to be distributedthroughout the whole mass; and after a certain interval of time, thestate of any part of the solid depends not upon the distribution of theinitial heat, but simply upon its quantity. The last result is not dueto the increase of the distance included between any point of the massand the part which has been heated; it is entirely due to the increaseof the time elapsed. In all problems submitted to analysis, the expo-nents are absolute numbers, and not quantities. We ought not to omitthe parts of these exponents which are incomparably smaller than theothers, but only those whose absolute values are extremely small . . 376

383—385. The same remarks apply to the distribution of heat in an infinitesolid 382

SECTION III.

THE HIGHEST TEMPERATURES IN AN INFINITE SOLID.

386, 387. The heat contained in part of the prism distributes itself through-out the whole mass. The temperature at a distant point rises pro-gressively, arrives at its greatest value, and then decreases. The time

TABLE OF CONTENTS. x x i

A R T * PAGE

after which this maximum occurs, is a function of the distance x.Expression of this function for a prism whose heated points have re-ceived the same initial temperature 335

388—391. Solution of a problem analogous to the foregoing. Differentresults of the solution ' 337

392—395. The movement of heat in an infinite solid is considered; andthe highest temperatures, at parts very distant from the part originallyheated, are determined 392

SECTION IV.

COMPAKISON OF THE INTEGRALS.

396. First integral (a) of the equation 37 = j i («)• This integral expresses

the movement of heat in a ring 396397. Second integral (/3) of the same equation (a). It expresses the linear

movement of heat in an infinite solid 398398. Two other forms (7) and (5) of the integral, which are derived, like the

preceding form, from the integral (a) n>t

399. 400. First development of the value of v according to increasing powersof the time t. Second development according to the powers of v. Thefirst must contain a single arbitrary function of t 899

401. Notation appropriate to the representation of these developments. Theanalysis which is derived from it dispenses with effecting the develop-ment in series 402

402. Application to the equations :

cVv d2v d2v , d2v d4v .+ ( ) nd + ° { d ) m

403. Application to the equations :

d2v dtv o d*v_

dv d2v , dAv deva n d dF = a ^ + i ^ + c r ^ + & C W • . . . 405

404. Use of the theorem E of Article 361, to form the integral of equation (/)of the preceding Article . 407

405. Use of the same theorem to form the integral of equation (d) whichbelongs to elastic plates 409

406. Second form of the same integral 412407. Lemmas which serve to effect these transformations . . . . 413408. The theorem expressed by equation (E)y Art. 361, applies to any number

of variables • . . . 415409. Use of this proposition to form the integral of equation (c) of Art. 402 . 416410. Application of the same theorem to the equation

d2v dH d2v n

dx2 dy1 dz2

XX11 TABLE OF CONTENTS.

ART, PAGE

411. Integral of equation (e) of vibrating elastic surfaces . . . . 4 1 9412. Second form of the integral 421413. Use of the same theorem to obtain the integrals, by summing the

series which represent them. Application to the equation

dv dH

Integral under finite form containing two arbitrary functions of t . . 422414. The expressions change form when we use other limits of the definite

integrals 425415. 416. Construction which serves to prove the general equation

417. Any limits a and b may be taken for the integral with respect to a.These limits are those of the values of x which correspond to existingvalues of the function f(x). Every other value of x gives a nul resultfor/(a?) 429

418, The same remark applies to the general equation

f(x) = ~ S f " + " f * daf(a) COS^" {x-a),

the second member of which represents a periodic function . . . 432419. The chief character of the theorem expressed by equation (B) consists

in this, that the sign / of the function is transferred to another unknowna, and that the chief variable x is only under the symbol cosine . . 433

420. Use of these theorems in the analysis of imaginary quantities . . 435d2v d2v

421. Application to the equation—2 + ^~2 = 0 436422. General expression of the fluxion of the order *,

*4^ 437dxi

423. Construction which serves to prove the general equation. Consequencesrelative to the extent of equations of this kind, to the values of / (x)which correspond to the limits of x, to the infinite values of f(x). . 438

424—427. The method which consists in determining by definite integralsthe unknown coefficients of the development of a function of a underthe form

() b () fo) &c,is derived from the elements of algebraic analysis. Example relative tothe distribution of heat in a solid sphere. By examining from thispoint of view the process which serves to determine the coefficients, wesolve easily problems which may arise on the employment of all the termsof the second member, on the discontinuity of functions, on singular orinfinite values. The equations which are obtained by this method ex-press either the variable state, or the initial state of masses of infinitedimensions. The form of the integrals which belong to the theory of

TABLE OF CONTENTS. xxiii

ART. PAGE

heat, represents at the same time the composition of simple movements,and that of an infinity of partial effects, due to the action of all points ofthe solid 441

428. General remarks on the method which has served to solve the analyticalproblems of the theory of heat . . . . . . . . . 450

429. General remarks on the principles from which we have derived the dif-ferential equations of the movement of heat 456

430. Terminology relative to the general properties of heat . . . . 462431. Notations proposed 463432. 433. General remarks on the nature of the coefficients which enter into

the differential equations of the movement of heat • . . » . 4 6 4

ERRATA.

Page 9, line 28, for III. read IV.Pages 54, 55, for k read K.Page 189, line 2, The equation should be denoted (A).Page 205, last line but one, for x read X,

Page 298, line 18, for -^ read ^ .

Page 299, line 16, for of read in.„ ,, last line, read

du <p (t sin u) = 7T0 + tSrf' + 4 #20" + &c-

Page 300, line 3, for A2, AA) A6i read wA2, TTA4, TT.4G

Page 407, line 12, for d<p read dp.Page 432, line 13, read (x-a).

fJo

PRELIMINARY DISCOURSE.

PRIMARY causes are unknown to us; but are subject to simpleand constant laws, which may be discovered by observation, thestudy of them being the object of natural philosophy.

Heat, like gravity, penetrates every substance of the universe,its rays occupy all parts of space. The object of our work is toset forth the mathematical laws which this element obeys. Thetheory of heat will hereafter form one of the most importantbranches of general physics.

The knowledge of rational mechanics, which the most ancientnations had been able to acquire, has not come down to us, andthe history of this science, if we except the first theorems inharmony, is not traced up beyond the discoveries of Archimedes.This great geometer explained the mathematical principles ofthe equilibrium of solids and fluids. About eighteen centurieselapsed before Galileo, the originator of dynamical theories, dis-covered the laws of motion of heavy bodies. Within this newscience Newton comprised the whole system of the universe. Thesuccessors of these philosophers have extended these theories, andgiven them an admirable perfection: they have taught us thatthe most diverse phenomena are subject to a small number offundamental laws which are reproduced in all the acts of nature.It is recognised that the same principles regulate all the move-ments of the stars, their form, the inequalities of their courses,the equilibrium and the oscillations of the seas, the harmonicvibrations of air and sonorous bodies, the transmission of light,capillary actions, the undulations of fluids, in fine the most com-plex effects of all the natural forces, and thus has the thought

F. H. 1

2 THEORY OF HEAT.

of Newton been confirmed: quod tarn paucis tarn multa prcestetgeometria gloriatur1.

But whatever may be the range of mechanical theories, theydo not apply to the effects of heat. These make up a specialorder of phenomena, which cannot be explained by the principlesof motion and equilibrium. We have for a long time been inpossession of ingenious instruments adapted to measure manyof these effects; valuable observations have been collected; butin this manner partial results only have become known, andnot the mathematical demonstration of the laws which includethem all.

I have deduced these laws from prolonged study and at-tentive comparison of the facts known up to this time: all thesefacts I have observed afresh in the course of several years withthe most exact instruments that have hitherto been used.

To found the theory, it was in the first place necessary todistinguish and define with precision the elementary propertieswhich determine the action of heat. I then perceived that all thephenomena which depend on this action resolve themselves intoa very small number of general and simple facts; whereby everyphysical problem of this kind is brought back to an investiga-tion of mathematical analysis. From these general facts I haveconcluded that to determine numerically the most varied move-ments of heat, it is sufficient to submit each substance to threefundamental observations. Different bodies in fact do not possessin the same degree the power to contain heat, to receive or transmitit across their surfaces, nor to conduct it through the interior oftheir masses. These are the three specific qualities which ourtheory clearly distinguishes and shews how to measure.

It is easy to judge how much these researches concern thephysical sciences and civil economy, and what may be theirinfluence on the progress of the arts which require the employ-ment and distribution of heat. They have also a necessary con-nection with the system of the world, and their relations becomeknown when we consider the grand phenomena which take placenear the surface of the terrestrial globe.

1 Philosophies naturalis principia mathematica. Auctoris praefatio ad lectorem.Ac gloriatur geometria quod tarn paucis principiis aliunde petitis tarn multapnestet. [A. F.]

PRELIMINARY DISCOURSE. 3

In fact, the radiation of the sun in which this planet isincessantly plunged, penetrates the air, the earth, and the waters;its elements are divided, change in direction every way, and,penetrating the mass of the globe, would raise its mean tem-perature more and more, if the heat acquired were not exactlybalanced by that which escapes in rays from all points of thesurface and expands through the sky.

Different climates, unequally exposed to the action of solarheat, have, after an immense time, acquired the temperatures•proper to their situation. This effect is modified by several ac-cessory causes, such as elevation, the form of the ground, theneighbourhood and extent of continents and seas, the state of thesurface, the direction of the winds.

The succession of day and night, the alternations of theseasons occasion in the solid earth periodic variations, which arerepeated every day or every year: but these changes becomeless and less sensible as the point at which they are measuredrecedes from the surface. No diurnal variation can be detectedat the depth of about three metres [ten feet]; and the annualvariations cease to be appreciable at a depth much less thansixty metres. The temperature at great depths is then sensiblyfixed at a given place: but it is not the same at all points of thesame meridian; in general it rises as the equator is approached.

The heat which the sun has communicated to the terrestrialglobe, and which has produced the diversity of climates, is nowsubject to a movement which has become uniform. It advanceswithin the interior of the mass which it penetrates throughout,and at the same time recedes from the plane of the equator, andproceeds to lose itself across the polar regions.

In the higher regions of the atmosphere the air is very rareand transparent, and retains but a minute part of the heat ofthe solar rays: this is the cause of the excessive cold of elevatedplaces. The lower layers, denser and more heated by the landand water, expand and rise up : they are cooled by the veryfact of expansion. The great movements of the air, such asthe trade winds which blow between the tropics, are not de-termined by the attractive forces of the moon and sun. Theaction of these celestial bodies produces scarcely perceptibleoscillations in a fluid so rare and at so great a distance. It

1—2

4 THEORY OF HEAT.

is the changes of temperature which periodically displace everypart of the atmosphere.

The waters of the ocean are differently exposed at theirsurface to the rays of the sun, and the bottom of the basinwhich contains them is heated very unequally from the polesto the equator. These two causes, ever present, and combinedwith gravity and the centrifugal force, keep up vast movementsin the interior of the seas. They displace and mingle all theparts, and produce those general and regular currents whichnavigators have noticed.

Radiant heat which escapes from the surface of all bodies,and traverses elastic media, or spaces void of air, has speciallaws, and occurs with widely varied phenomena. The physicalexplanation of many of these facts is already known ; the mathe-matical theory which I have formed gives an exact measure ofthem. It consists, in a manner, in a new catoptrics whichhas its own theorems, and serves to determine by analysis allthe effects of heat direct or reflected.

The enumeration of the chief objects of the theory sufficientlyshews the nature of the questions which I have proposed tomyself. What are the elementary properties which it is requisiteto observe in each substance, and what are the experimentsmost suitable to determine them exactly? If the distributionof heat in solid matter is regulated by constant laws, what isthe mathematical expression of those laws, and by what.analysismay we derive from this expression the complete solution ofthe principal problems ? Why do terrestrial temperatures ceaseto be variable at a depth so small with respect to the radiusof the earth ? Every inequality in the movement of this planetnecessarily occasioning an oscillation of the solar heat beneaththe surface, what relation is there between the duration of itsperiod, and the depth at which the temperatures become con-stant ?

What time must have elapsed before the climates could acquirethe different temperatures which they now maintain; and whatare the different causes which can now vary their mean heat?"Why do not the annual changes alone in the distance of thesun from the earth, produce at the surface of the earth veryconsiderable changes in the temperatures ?


From what characteristic can we ascertain that the earthhas not entirely lost its original heat; and what are the exactlaws of the loss ?

If, as several observations indicate, this fundamental heatis not wholly dissipated, it must be immense at great depths,and nevertheless it has no sensible influence at the present timeon the mean temperature of the climates. The effects whichare observed in them are due to the action of the solar rays.But independently of these two sources of heat, the one funda-mental and primitive, proper to the terrestrial globe, the other dueto the presence of the sun, is there not a more universal cause,which determines the temperature of the heavens, in that partof space which the solar system now occupies? Since the ob-served facts necessitate this cause, what are the consequencesof an exact theory in this entirely new question; how shall webe able to determine that constant value of the temperature ofspace, and deduce from it the temperature which belongs to eachplanet ?

To these questions must be added others which depend onthe properties of radiant heat. The physical cause of the re-flection of cold, that is to say the reflection of a lesser degreeof heat, is very distinctly known; but what is the mathematicalexpression of this effect ?

On what general principles do the atmospheric temperaturesdepend, whether the thermometer which measures them receivesthe solar rays directly, on a surface metallic or unpolished,or whether this instrument remains exposed, during the night,under a sky free from clouds, to contact with the air, to radiationfrom terrestrial bodies, and to that from the most distant andcoldest parts of the atmosphere ?

The intensity of the rays which escape from a point on thesurface of any heated body varying with their inclination ac-cording to a law which experiments have indicated, is there not anecessary mathematical relation between this law and the generalfact of the equilibrium of heat; and what is the physical cause ofthis inequality in intensity ?

Lastly, when heat penetrates fluid masses, and determines inthem internal movements by continual changes of the temperatureand density of each molecule, can we still express, by differential

6 THEORY OF HEAT.

equations, the laws of such a compound effect; and what is theresulting change in the general equations of hydrodynamics ?

Such are the chief problems which I have solved, and whichhave never yet been submitted to calculation. If we considerfurther the manifold relations of this mathematical theory tocivil uses and the technical arts, we shall recognize completelythe extent of its applications. It is evident that it includes anentire series of distinct phenomena, and that the study of itcannot be omitted without losing a notable part of the science ofnature.^ The principles of the theory are derived, as are those ofrational mechanics, from a very small number of primary facts,the causes of which are not considered^ by geometers, but whichthey admit as the results of common observations confirmed by allexperiment.

The differential equations of the propagation of heat expressthe most general conditions, and reduce the physical questions toproblems of pure analysis, and this is the proper object of theory.They are not less rigorously established than the general equationsof equilibrium and motion. In order to make this comparisonmore perceptible, we have always preferred demonstrations ana-logous to those of the theorems which serve as the foundationof statics and dynamics. These equations still exist, but receivea different form, when they express the distribution of luminousheat in transparent bodies, or the movements which the changesof temperature and density occasion in the interior of fluids.The coefficients which they contain are subject to variations whoseexact measure is not yet known; but in all the natural problemswhich it most concerns us to consider, the limits of temperaturediffer so little that we may omit the variations of these co-efficients.

The equations of the movement of heat, like those whichexpress the vibrations of sonorous bodies, or the ultimate oscilla-tions of liquids, belong to one of the most recently discoveredbranches of analysis, which it is very important to perfect. Afterhaving established these differential equations their integrals mustbe obtained; this process consists in passing from a commonexpression to a particular solution subject to all the given con-ditions. This difficult investigation requires a special analysis


founded on new theorems, whose object we could not in thisplace make - known. The method which is derived from themleaves nothing vague and indeterminate in the solutions, it leadsthem up to the final numerical applications, a necessary conditionof every investigation, without which we should only arrive atuseless transformations.

The same theorems which have made known to us theequations of the movement of heat, apply directly to certain pro-blems of general analysis and dynamics whose solution has for along time been desired.

Profound study of nature is the most fertile source of mathe-matical discoveries. Not only has this study, in offering a de-terminate object to investigation, the advantage of excludingvague questions and calculations without issue; it is besides asure method of forming analysis itself, and of discovering theelements which it concerns us to know, and which natural scienceought always to preserve: these are the fundamental elementswhich are reproduced in all natural effects.

We see, for example, that the same expression whose abstractproperties geometers had considered, and which in this respectbelongs to general analysis, represents as well the motion of lightin the atmosphere, as it determines the laws of diffusion of heatin solid matter, and enters into all the chief problems of thetheory of probability.

The analytical equations, unknown to the ancient geometers,which Descartes was the first to introduce into the study of curvesand surfaces, are not restricted to the properties of figures, and tothose properties which are the object of rational mechanics; theyextend to all general phenomena. There cannot be a languagemore universal and more simple, more free from errors and fromobscurities, that is to say more worthy to express the invariablerelations of natural things.

Considered from this point of view, mathematical analysis is asextensive as nature itself; it defines all perceptible relations,measures times, spaces, forces, temperatures; this difficult scienceis formed slowly, but it preserves every principle which it has onceacquired; it grows and strengthens itself incessantly in the midstof the many variations and errors of the human mind.

Its chief attribute is clearness; it has no marks to express con-

g THEORY OF HEAT.

fused notions. It brings together phenomena the most diverse,and discovers the hidden analogies which unite them. If matterescapes us, as that of air and light, by its extreme tenuity, ifbodies are placed far from us in the immensity of space, if manwishes to know the aspect of the heavens at successive epochsseparated by a great number of centuries, if the actions of gravityand of heat are exerted in the interior of the earth at depthswhich will be always inaccessible, mathematical analysis can yetlay hold of the laws of these phenomena'. It makes them presentand measurable, and seems to be a faculty of the human minddestined to supplement the shortness of life and the imperfec-tion of the senses; and what is still more remarkable, it followsthe same course in the study of all phenomena; it interprets themby the same language, as if to attest the unity and simplicity ofthe plan of the universe, and to make still more evident thatunchangeable order which presides over all natural causes.

The problems of the theory of heat present so many examplesof the simple and constant dispositions which spring from thegeneral laws of nature ; and if the order which is established inthese phenomena could be grasped by our senses, it would producein us an impression comparable to the sensation of musical sound.

The forms of bodies are infinitely varied ; the distribution ofthe heat which penetrates them seems to be arbitrary and confused;but all the inequalities are rapidly cancelled and disappear as timepasses on. The progress of the phenomenon becomes more regularand simpler, remains finally subject to a definite law which is thesame in all cases, and which bears no sensible impress of the initialarrangement.

All observation confirms these consequences. The analysisfrom which they are derived separates and expresses clearly, 1° thegeneral conditions, that is to say those which spring from thenatural properties of heat, 2° the effect, accidental but continued,of the form or state of the surfaces ; 3° the effect, not permanent,of the primitive distribution.

In this work we have demonstrated all the principles of thetheory of heat, and solved all the fundamental problems. Theycould have been explained more concisely by omitting the simplerproblems, and presenting in the first instance the most generalresults; but we wished to shew the actual origin of the theory and


its gradual progress. When this knowledge has been acquiredand the principles thoroughly fixed, it is preferable to employ atonce the most extended analytical methods, as we have done inthe later investigations. This is also the course which we shallhereafter follow in the memoirs which will be added to this work,and which will form in some manner its complement *; and by thismeans we shall have reconciled, so far as it can depend on our-selves, the necessary development of principles with the precisionwhich becomes the applications of analysis.

The subjects of these memoirs will be, the theory of radiantheat, the problem of the terrestrial temperatures, that of thetemperature of dwellings, the comparison of theoretic results withthose which we have observed in different experiments, lastly thedemonstrations of the differential equations of the movement ofheat in fluids.

The work which we now publish has been written a long timesince; different circumstances have delayed and often interruptedthe printing of it. In this interval, science has been enriched byimportant observations; the principles of our analysis, which hadnot at first been grasped, have become better known; the resultswhich we had deduced from them have been discussed and con-firmed. We ourselves have applied these principles to newproblems, and have changed the form of some of the proofs.The delays of publication will have contributed to make the workclearer and more complete.

The subject of our first analytical investigations on the transferof heat was its distribution amongst separated masses; these havebeen preserved in Chapter III., Section II. The problems relativeto continuous bodies, which form the theory rightly so called, weresolved many years afterwards ; this theory was explained for thefirst time in a manuscript work forwarded to the Institute ofFrance at the end of the year 1807, an extract from which waspublished in the Bulletin des Sciences (Society Philomatique, year1808, page 112). We added to this memoir, and successively for-warded very extensive notes, concerning the convergence of series,the diffusion of heat in an infinite prism, its emission in spaces

1 These memoirs were never collectively published as a sequel or complementto the Theorie Analytique de la Chaleur. But, as will be seen presently, the authorhad written most of them before the publication of that work in 1822. [A. ¥.]

10 THEORY OF HEAT.

void of air, the constructions suitable for exhibiting the chieftheorems, and the analysis of the periodic movement at the sur-face of the earth. Our second memoir, on the propagation ofheat, was deposited in the archives of the Institute, on the 28th ofSeptember, 1811. It was formed out of the preceding memoir andthe notes already sent in; the geometrical constructions andthose details of analysis which had no necessary relation to thephysical problem were omitted, and to it was added the generalequation which expresses the state of the surface. This secondwork was sent to press in the course of 1821, to be inserted inthe collection of the Academy of Sciences. It is printed withoutany change or addition ; the text agrees literally with the depositedmanuscript, which forms part of the archives of the Institute \

In this memoir, and in the writings which preceded it, will befound a first explanation of applications which our actual work

1 It appears as a memoir and supplement in volumes IV. and V. of the Me-moires de VAcademie des Sciences. For convenience of comparison with the tableof contents of the Analytical Theory of Heat, we subjoin the titles and heads ofthe chapters of the printed memoir:

THEORIE DU MOUVBMENT DE LA CHALEUR DANS LES CORPS SOLIDES, PAR M.

FOURIER. [Memoires de VAcademie Boy ale des Sciences de VInstitut de France.Tome IV. (for year 1819). Paris 1821]I. Exposition.II. Notions generates et definitions preliminaires.III. Equations du mouvement de la chaleur.IV. Du mouvement lineaire et varie de la chaleur dans une armille.V. De la propagation de la chaleur dans une lame rectangulaire dont les temperatures

sont constantes.

VI. De la communication de la chaleur entre des masses disjointes.VII. Du mouvement varie de la chaleur dans une sphere solide.VIII. Du mouvement varie de la chaleur dans un cylindre solide.IX. De la propagation de la chaleur dans un prisme dont Vextremite est assujcttie

a une temperature const ante.X. Du mouvement varie de la chaleur dans un solide de forme cubique.XI. Du mouvement lineaire et varii de la chaleur dans les corps dont une dimension

est infinie.

SUITE DU MEMOIRE INTITULE : THEORIE DU MOUVEMENT DE LA CHALEUR DANS

LES CORPS SOLIDES ; PAR M. FOURIER. [Memoires de VAcademie Eoyale des Sciencesde VInstitut de France. Tome V. (for year 1820). Paris, 1826.]XII. Des temperatures terrestres, et du mouvement de la chaleur dans Vintericur

d'une sphere solide, dont la surface est assujettie a des changemens periodiquesde temperature.

XIII. Des lois mathematiques de Vequilibre de la chaleur rayonnante.XIV. Comparaison des resultats de la theorie avec ceux de diverses experiences

[A. P.]

PRELIMINARY DISCOURSE. H

does not contain; they will be treated in the subsequent memoirs1

at greater length, and, if it be in our power, with greater clear-ness. The results of our labours concerning the same problemsare also indicated in several articles already published. Theextract inserted in the Annales de Ghimie et de Physique shewsthe aggregate of our researches (Vol. in. page 350, year 1816).We published in the Annales two separate notes, concerningradiant heat (Vol. IV. page 128, year 1817, and Vol. VI. page 259,year 1817).

Several other articles of the same collection present the mostconstant results of theory and observation; the utility and theextent of thermological knowledge could not be better appreciatedthan by the celebrated editors of the Annales2.

In the Bulletin des Sciences (SocUti philomatique year 1818,page 1, and year 1820, page 60) will be found an extract froma memoir on the constant or variable temperature of dwellings,and an explanation of the chief consequences of our analysis ofthe terrestrial temperatures.

M. Alexandre de Humboldt, whose researches embrace all thegreat problems of natural philosophy, has considered the obser-vations of the temperatures proper to the different climatesfrom a novel and very important point of view (Memoir on Iso-thermal lines, Societe d'Arcueil, Vol. in. page 462); (Memoir onthe inferior limit of perpetual snow, Annales de Ghimie et deFhysique, Vol. Y. page 102, year 1817).

As to the differential equations of the movement of heat influids3 mention has been made of them in the annual history ofthe Academy of Sciences. The extract from our memoir shewsclearly its object and principle. (Analyse des travaux de VAca-dhnie des Sciences, by M. De Lambre, year 1820.)

The examination of the repulsive forces produced by heat,which determine the statical properties of gases, does not belong

1 See note, page 9, and the notes, pages 11—13.2 Gay-Lussac and Arago. See note, p. 13.3 Memoires de VAcademie des Sciences, Tome XII.,, Paris, 1833, contain on pp.

507—514, Memoir-e a"analyse sur le mouvement de la chaleur dans les fluides, par 31.Fourier. Lu a VAcademie Eoyale des Sciences, 4 Sep. 1820. It is followed on pp.515 530 by Extrait des notes manuscrites conservees par Vauteur. The memoiris signed Jh. Fourier, Paris, 1 Sep. 1820, but was published after the death of theauthor. [A. F.]

12 THEORY OF HEAT.

to the analytical subject which we have considered. This questionconnected with the theory of radiant heat has just been discussedby the illustrious author of the Mecanique celeste, to whom allthe chief branches of mathematical analysis owe importantdiscoveries. (Connaissance des Temps, years 1824-5.)

The new theories explained in our work are united for everto the mathematical sciences, and rest like them on invariablefoundations; all the elements which they at present possess theywill preserve, and will continually acquire greater extent. Instru-ments will be perfected and experiments multiplied. The analysiswhich we have formed will be deduced from more general, thatis to say, more simple and more fertile methods common to manyclasses of phenomena. For all substances, solid or liquid, forvapours and permanent gases, determinations will be made of allthe specific qualities relating to heat, and of the variations of thecoefficients which express them1. At different stations on theearth observations will be made, of the temperatures of theground at different depths, of the intensity of the solar heat andits effects, constant or variable, in the atmosphere, in the oceanand in lakes; and the constant temperature of the heavens properto the planetary regions will become known2. The theory itself

1 Memoires de VAcademic des Sciences, Tome VIII., Paris 1829, contain onpp. 581—622, Memoir'e sur la Theorie Analytique de la Chaleur, par M. Fourier.This was published whilst the author was Perpetual Secretary to the Academy.The first only of four parts of the memoir is printed. The contents of all arestated. I. Determines the temperature at any point of a prism whose terminaltemperatures are functions of the time, the initial temperature at any point beinga function of its distance from one end. II . Examines the chief consequences ofthe general solution, and applies it to two distinct cases, according as the tempe-ratures of the ends of the heated prism are periodic or not. III. Is historical,enumerates the earlier experimental and analytical researches of other writersrelative to the theory of heat; considers the nature of the transcendental equations• appearing in the theory; remarks on the employment of arbitrary functions;replies to the objections of M. Poisson; adds some remarks on a problem of themotion of waves. IV. Extends the application of the theory of heat by takingaccount, in the analysis, of variations in the specific coefficients which measurethe capacity of substances for heat, the permeability of solids, and the penetra-bility of their surfaces. [A. F.]

2 Memoires de VAcademie des Sciences, Tome VII., Pans, 1827, contain onpp. 569—604, Memoire sur les temperatures du globe terrestre et des espaces plane-taires, par M. Fourier. The memoir is entirely descriptive; it was read before theAcademy, 20 and 29 Sep. 1824 (Annales de Chimie et de Physique, 1824 xxvnp. 136). [A. F.]


will direct all these measures, and assign their precision. Noconsiderable progress can hereafter be made which is not foundedon experiments such as these; for mathematical analysis candeduce from general and simple phenomena the expression of thelaws of nature; but the special application of these laws to very-complex effects demands a long series of exact observations.

The complete list of the Articles on Heat, published by M. Fourier, in theAnnales de Chimie et de Physique, Series 2, is as follows :

1816. III. pp. 350—375. Theorie de la Chaleur (Extrait). Description by theauthor of the 4to volume afterwards published in 1822 without the chapters onradiant heat, solar heat as it affects the earth, the comparison of analysis withexperiment, and the history of the rise and progress of the theory of heat.

1817. IV. pp. 128—145. Note sur la Chaleur rayonnante. Mathematicalsketch on the sine law of emission of heat from a surface. Proves the author'sparadox on the hypothesis of equal intensity of emission in all directions.

1817. VI. pp. 259—303. Questions sur la theorie physique de la chaleurrayonnante. An elegant physical treatise on the discoveries of Newton, Pictet,Wells, Wollaston, Leslie and Prevost.

1820. XIII. pp. 418—438. Sur le refroidissement seculaire de la terre (Extrait),Sketch of a memoir, mathematical and descriptive, on the waste of the earth'sinitial heat.

1824. XXVII. pp. 136—167. Eemarques generates sur les temperatures du globeterrestre et des espaces planetaires. This is the descriptive memoir referred toabove, Mem. Acad. d. Sc. Tome VII.

1824. XXVII. pp. 236—281. Resume theorique des proprietes de la chaleurrayonnante. Elementary analytical account of surface-emission and absorptionbased on the principle of equilibrium of temperature.

1825. XXVIII. pp. 337—365. Bemarques sur la theorie mathematique de lachaleur rayonnante. Elementary analysis of emission, absorption and reflectionby walls of enclosure uniformly heated. At p. 364, M. Fourier promises a Theoriephysique de la chaleur to contain the applications of the Theorie Analytiqueomitted from the work published in 1822.

1828. XXXVII. pp. 291—315. Recherches experimentales sur la faculte con-ductrice des corps minces soumis a faction de la chaleur^ et description dhtn nouveauthermametre de contact. A thermoscope of contact intended for lecture demonstra-tions is also described. M. Emile Verdet in his Conferences de Physique, Paris,1872. Part I. p. 22, has stated the practical reasons against relying on thetheoretical indications of the thermometer of contact. [A. F.]

Of the three notices of memoirs by M. Fourier, contained in the Bulletin desSciences par la Societe Philomatique, and quoted here at pages 9 and 11, the firstwas written by M. Poisson, the mathematical editor of the Bulletin, the other two byM. Fourier. [A. F.]

THEORY OF HEAT.

FA ignem regunt numeri.—PLATO1.

CHAPTER I.

INTRODUCTION.

FIRST SECTION.

Statement of the Object of the Work

1. THE effects of heat are subject to constant laws whichcannot be discovered without the aid of mathematical analysis.The object of the theory which we are about to explain is todemonstrate these laws; it reduces all physical researches onthe propagation of heat, to problems of the integral calculuswhose elements are given by experiment. No subject has moreextensive relations with the progress of industry and the naturalsciences; for the action of heat is always present, it penetratesall bodies and spaces, it influences the processes of the arts,and occurs in all the phenomena of the universe.

When heat is unequally distributed among the different partsof a solid mass, it tends to attain equilibrium, and passes slowlyfrom the parts which are more heated to those which are less;and at the same time it is dissipated at the surface, and lostin the medium or in the void. The tendency to uniform dis-tribution and the spontaneous emission which acts at the surfaceof bodies, change continually the temperature at their differentpoints. The problem of the propagation of heat consists in

1 Cf. Plato, Timaus, 53, B.

Sre 5' eirexeLpelro KO a/Mia Oat TO vdv, Tvp Trpurov KOX yrju /cat depa Kal vdwp-xwcLTio-aTo .[6 6ecs] ddeal re /cat dpidfxo?s. [A. F.]

CH. I. SECT. I.] INTRODUCTION. 15

determining what is the temperature at each point of a bodyat a given instant, supposing that the initial temperatures areknown. The following examples will more clearly make knownthe nature of these problems.

2. If we expose to the continued and uniform action of asource of heat, the same part of a metallic ring, whose diameteris large, the molecules nearest to the source will be first heated,and, after a certain time, every point of the solid will haveacquired very nearly the highest temperature which it can attain.This limit or greatest temperature is not the same at differentpoints; it becomes less and less according as they become moredistant from that point at which the source of heat is directlyapplied.

When the temperatures have become permanent, the sourceof heat supplies, at each instant, a quantity of heat which exactlycompensates for that which is dissipated at all the points of theexternal surface of the ring.

If now the source be suppressed, heat will continue to bepropagated in the interior of the solid, but that which is lostin the medium or the void, will no longer be compensated asformerly by the supply from the source, so that all the tempe-ratures will vary and diminish incessantly until they have be-come equal to the temperatures of the surrounding medium.

3. Whilst the temperatures are permanent and the sourceremains, if at every point of the mean circumference of the rino-an ordinate be raised perpendicular to the plane of the ring,whose length is proportional to the fixed temperature at thatpoint, the curved line which passes through the ends of theseordinates will represent the permanent state of the temperatures,and it is very easy to determine by analysis the nature of thisline. It is to be remarked that the thickness of the ring issupposed to be sufficiently small for the temperature to besensibly equal at all points of the same section perpendicularto the mean circumference. When the source is removed, theline which bounds the ordinates proportional to the temperaturesat the different points will change its form continually. Theproblem consists in expressing, by one equation, the variable

16 THEOKY OF HEAT. [CHAP. I.

form of this curve, and in thus including in a single formulaall the successive states of the solid.

4. Let z be the constant temperature at a point m of themean circumference, x the distance of this point from the source,that is to say the length of the arc of the mean circumference,included between the point m and the point o wdiich correspondsto the position of the source; z is the highest temperaturewhich the point m.can attain by virtue of the constant actionof the source, and this permanent temperature z is a functionf{x) of the distance x. The first part of the problem consistsin determining the function f(x) which represents the permanentstate of the solid.

Consider next the variable state which succeeds to the formerstate as soon as the source has been removed ; denote by t thetime which has passed since the suppression of the source, andby v the value of the temperature at the point m after thetime t. The quantity v will be a certain function F (x, t) ofthe distance x and the time t; the object of the problem is todiscover this function F (x, t), of which we only know as yetthat the initial value is f (x)9 so that we ought to have theequation / (.r) = F (x, o).

5. If we place a solid homogeneous mass, having the formof a sphere or cube, in a medium maintained at a constant tem-perature, and if it remains immersed for a very long time, it willacquire at all its points a temperature differing very little fromthat of the fluid. Suppose the mass to be withdrawn in orderto transfer it to a cooler medium, heat will begin to be dissi-pated at its surface; the temperatures at different points of themass will not be sensibly the same, and if we suppose it dividedinto an infinity of layers by surfaces parallel to its external sur-face, each of those layers will transmit, at each instant, a certainquantity of heat to the layer which surrounds it. If it beimagined that each molecule carries a separate thermometer,which indicates its temperature at every instant, the state ofthe solid will from time to time be represented by the variablesystem of all these thermometric heights. It is required toexpress the successive states by analytical formula?, so that we

SECT. I.] INTRODUCTION. 17

may know at any given instant the temperatures indicated byeach thermometer, and compare the quantities of heat whichflow during the same instant, between two adjacent layers, orinto the surrounding medium.

6. If the mass is spherical, and we denote by x the distanceof a point of this mass from the centre of the sphere, by t thetime which has elapsed since the commencement of the cooling,and by v the variable temperature of the point m, it is easy to seethat all points situated at the same distance x from the centreof the sphere have the same temperature v. This quantity v is acertain function F (x, t) of the radius x and of the time t; it mustbe such that it becomes constant whatever be the value of x, whenwe suppose t to be nothing ; for by hypothesis, the temperature atall points is the same at the moment of emersion. The problemconsists in determining that function of x and t which expressesthe value of v.

7. In the next place it is to be remarked, that during thecooling, a certain quantity of heat escapes, at each instant, throughthe external surface, and passes into the medium. The value ofthis quantity is not constant; it is greatest at the beginning of thecooling. If however we consider the variable state of the internalspherical surface whose radius is x, we easily see that there mustbe at each instant a certain quantity of heat which traverses thatsurface, and passes through that part of the mass which is moredistant from the centre. This continuous flow of heat is variablelike that through the external surface, and both are quantitiescomparable with each other; their ratios are numbers whose vary-ing values are functions of the distance x, and of the time t whichhas elapsed. It is required to determine these functions.

8. If the mass, which has been heated by a long immersion ina medium, and whose rate of cooling we wish to calculate, isof cubical form, and if we determine the position of each point m bythree rectangular co-ordinates x, y, z, taking for origin the centreof the cube, and for axes lines perpendicular to the faces, we seethat the temperature v of the point m after the time t, is a func-tion of the four variables x, y, z, and t The quantities ^ of heat

F. H.

18 THEORY OF HEAT. [CHAP. I.

which flow out at each instant through the whole external surfaceof the solid, are variable and comparable with each other; theirratios are analytical functions depending on the time t, the expres-sion of which must be assigned.

9. Let us examine also the case in which a rectangular prismof sufficiently great thickness and of infinite length, being sub-mitted at its extremity to a constant temperature, whilst the air,which surrounds it is maintained at a less temperature, has at lastarrived at a fixed state which it is required to determine. All thepoints of the extreme section at the base of the prism have, byhypothesis, a common and permanent temperature. I t is not thesame with a section distant from the source of heat; each of thepoints of this rectangular surface parallel to the base has acquireda fixed temperature, but this is not the same at different points ofthe same section, and must be less at points nearer to the surfaceexposed to the air. We see also that, at each instant, there flowsacross a given section a certain quantity of heat, which alwaysremains the same, since the state of the solid has become constant.The problem consists in determining the permanent temperatureat any given point of the solid, and the whole quantity of heatwhich, in a definite time, flows across a section whose position isgiven.

10. Take as origin of co-ordinates x, y, zy the centre of thebase of the prism, and as rectangular axes, the axis of the prismitself, and the two perpendiculars on the sides: the permanenttemperature v of the point m, whose co-ordinates are x} y, z, isa function of three variables F (#, y, z): it has by hypothesis aconstant value, when we suppose x nothing, whatever be the valuesof y and z. Suppose we take for the unit of heat that quantitywhich in the unit of time would emerge from an area equal to aunit of surface, if the heated mass which that area bounds, andwhich is formed of the same substance as the prism, were continu-ally maintained at the temperature of boiling water, and immersedin atmospheric air maintained at the temperature of melting ice.

We see that the quantity of heat which, in the permanentstate of the rectangular prism, flows, during a unit of time, acrossa certain section perpendicular to the axis, has a determinate ratio

SECT. I.] INTRODUCTION. 19

to the quantity of heat taken as unit. This ratio is not the samefor all sections: it is a function <£ (#) of the distance oc, at whichthe section is situated. It is required to find an analytical expres-sion of the function <£ Qv).

11. The foregoing examples suffice to give an exact idea ofthe different problems which we have discussed.

The solution of these problems has made us understand thatthe effects of the propagation of heat depend in the case of everysolid substance, on three elementary qualities, which are, its capa-city for heat, its own conducibility, and th£ exterior conducibility.

It has been observed that if two bodies of the same volumeand of different nature have equal temperatures, and if the samequantity of heat be added to them, the increments of temperatureare not the same; the ratio of these increments is the ratio oftheir capacities for heat. In this manner, the first of the threespecific elements which regulate the action of heat is exactlydefined, and physicists have for a long time known several methodsof determining its value. It is not the same with the two others;their effects have often been observed, but there is but one exacttheory which can fairly distinguish, define, and measure themwith precision.

The proper or interior conducibility of a body expresses thefacility with which heat is propagated in passing from one internalmolecule to another. The external or relative conducibility of asolid body depends on the facility with which heat penetrates thesurface, and passes from this body into a given medium, or passesfrom the medium into the solid. The last property is modified bythe more or less polished state of the surface ; it varies also accord-ing to the medium in which the body is immersed ; but theinterior conducibility can change only with the nature of thesolid.

These three elementary qualities are represented in ourformulae by constant numbers, and the theory itself indicatesexperiments suitable for measuring their valijes. As soon as theyare determined, all the problems relating to the propagation ofheat depend only on numerical analysis. The knowledge of thesespecific properties may be directly useful in several applications ofthe physical sciences; it is besides an element in the study and

2—2


description of different substances. It is a very imperfect know-ledge of bodies which ignores the relations which they have withone of the chief agents of nature. In general, there is no mathe-matical theory which has a closer relation than this with publiceconomy, since it serves to give clearness and perfection to thepractice of the numerous arts which are founded on the employ-ment of heat.

12. The problem of the terrestrial temperatures presentsone of the most beautiful applications of the theory of heat; thegeneral idea to be formed of it is this. Different parts of thesurface of the globe are unequally exposed to the influence of thesolar rays; the intensity of their action depends on the latitude ofthe place; it changes also in the course of the day and in thecourse of the year, and is subject to other less perceptible in-equalities. It is evident that, between the variable state of thesurface and that of the internal temperatures, a necessary relationexists, which may be derived from theory. We know that, at acertain depth below the surface of the earth, the temperature at agiven place experiences no annual variation: this permanentunderground temperature becomes less and less according as theplace is more and more distant from the equator. We may thenleave out of consideration the exterior envelope, the thickness ofwhich is incomparably small with respect to the earth's radius,and regard our planet as a nearly spherical mass, whose surfaceis subject to a temperature which remains constant at all pointson a given parallel, but is not the same on another parallel. Itfollows from this that every internal molecule has also a fixed tem-perature determined by its position. The mathematical problemconsists in discovering the fixed temperature at any given point,and the law which the solar heat follows whilst penetrating theinterior of the earth.

This diversity of temperature interests us still more, if weconsider the changes which succeed each other in the envelopeitself on the surface of which we dwell. Those alternations ofheat and cold which are reproduced everyday and in the course ofevery year, have been up to the present time the object of repeatedobservations. These we can now submit to calculation, and froma common theory derive all the particular facts which experience

S E C T - L ] INTRODUCTION. 21

has taught us. The problem is reducible to the hypothesis thatevery point of a vast sphere is affected by periodic temperatures ;analysis then tells us according to what law the intensity of thesevariations decreases according as the depth increases, what is theamount of the annual or diurnal changes at a given depth, theepoch of the changes, and how the fixed value of the undergroundtemperature is deduced from the variable temperatures observedat the surface.

13. The general equations of the propagation of heat arepartial differential equations, and though their form is very simplethe known methods1 do not furnish any general mode of integrat-ing them; we could not therefore deduce from them the valuesof the temperatures after a definite time. The numerical inter-pretation of the results of analysis is however necessary, and itis a degree of perfection which it would be very important to giveto every application of analysis to the natural sciences. So longas it is not obtained, the solutions may be said to remain in-complete and useless, and the truth which it is proposed todiscover is no less hidden in the formulao of analysis than it wasin the physical problem itself. We have applied ourselves withmuch care to this purpose, and we have been able to overcomethe difficulty in all the problems of which we have treated, andwhich contain the chief elements of the theory of heat. There isnot one of the problems whose solution does not provide conve-nient and exact means for discovering the numerical values of thetemperatures acquired, or those of the quantities of heat which

1 For the modern treatment of these equations consultPartielle Differentialgleichungen, von B. Riemann, Braunschweig, 2nd Ed., 1876.

The fourth section, Bewegung der Wdrme in festen Kb'rpern.Cours de physique mathematique, par E. Matthieu, Paris, 1873. The parts

relative to the differential equations of the theory of heat.The Functions of Laplace, Lame, and Bessel, by I. Todhunter, London, 1875.

Chapters XXI. XXV.—XXIX. which give some of Lamp's methods.Conferences de Physique, par E. Verdet, Paris, 1872 [(Euvres, Vol. iv. Part i.].

Legons sur la propagation de la chaleur par conductibilite. These are followed bya very extensive bibliography of the whole subject of conduction of heat.

For an interesting sketch and application of Fourier's Theory seeTheory of Heat, by Prof. Maxwell,1jon&<m, 1875 [4th Edition]. Chapter XVIII.

On the diffusion of heat by conduction.Natural Philosophy, by Sir W. Thomson and Prof Tait, Vol. i. Oxford, 1867.

Chapter VII. Appendix D, On the secular cooling of the earth. [A. F.]


have flowed through, when the values of the time and of thevariable coordinates are known. Thus will be given not only thedifferential equations which the functions that express the valuesof the temperatures must satisfy; but the functions themselveswill be given under a form which facilitates the numericalapplications.

14. In order that these solutions might be general, and havean extent equal to that of the problem, it was requisite that theyshould accord with the initial state of the temperatures, which isarbitrary. The examination of this condition "shews that we maydevelop in convergent series, or express by definite integrals,functions which are not subject to a constant law, and whichrepresent the ordinates of irregular or discontinuous lines. Thisproperty throws a new light on the theory of partial differen-tial equations, and extends the employment of arbitrary functionsby submitting them to the ordinary processes of analysis.

15. It still remained to compare the facts with theory. Withthis view, varied and exact experiments were undertaken, whoseresults were in conformity with those of analysis, and gave theman authority which one would have been disposed to refuse tothem in a new matter which seemed subject to so much uncer-tainty. These experiments confirm the principle from which westarted, and which is adopted by all physicists in spite of thediversity of their hypotheses on the nature of heat.

16. Equilibrium of temperature is effected not only by wayof contact, it is established also between bodies separated fromeach other, which are situated for a long time in the same region.This effect is independent of contact with a medium; we haveobserved it in spaces wholly void of air. To complete our theoryit was necessary to examine the laws which radiant heat follows,on leaving the surface of a body. It results from the observationsof many physicists and from our own experiments, that the inten-sities of the different rays, which escape in all directions from anypoint in the surface of a heated body, depend on the angles whichtheir directions make with the surface at the same point. Wehave proved that the intensity of a ray diminishes as the ray

S E C T - I.] INTRODUCTION. 23

makes a smaller angle with the element of surface, and that it isproportional to the sine of that angle1. This general law ofemission of heat which different observations had already indi-cated, is a necessary consequence of the principle of the equilibriumof temperature and of the laws of propagation of heat in solidbodies.

Such are the chief problems which have been discussed inthis work; they are all directed to one object only, that is toestablish clearly the mathematical principles of the theory of heat,and to keep up in this way with the progress of the useful arts,and of the study of nature.

17. From what precedes it is evident that a very extensiveclass of phenomena exists, not produced by mechanical forces, butresulting simply from the presence and accumulation of heat.This part of natural philosophy cannot be connected with dy-namical theories, it has principles peculiar to itself, and is foundedon a method similar to that of other exact sciences. The solarheat, for example, which penetrates the interior of the globe, dis-tributes itself therein according to a regular law which does notdepend on the laws of motion, and cannot be determined by theprinciples of mechanics. The dilatations which the repulsiveforce of heat produces, observation of which serves to measuretemperatures, are in truth dynamical effects; but it is not thesedilatations which we calculate, when we investigate the laws ofthe propagation of heat.

18. There are other more complex natural effects, whichdepend at the same time on the influence of heat, and of attrac-tive forces: thus, the variations of temperatures which the move-ments of the sun occasion in the atmosphere and in the ocean,change continually the density of the different parts of the airand the waters. The effect of the forces which these masses obeyis modified at every instant by a new distribution of heat, andit cannot be doubted that this cause produces the regular winds,and the chief currents of the sea; the solar and lunar attractionsoccasioning in the atmosphere effects but slightly sensible, andnot general displacements. It was therefore necessary, in order to

i Mem. Acad. d. Sc. Tome V. Paris, 1826, pp. 179—213. [A. F.]

2t THEORY OF HEAT. [CHAP. I.

submit these grand phenomena to calculation, to discover themathematical laws of the propagation of heat in the interior ofmasses.

19. It will be perceived, on reading this work, that heat at-tains in bodies a regular disposition independent of the originaldistribution, which may be regarded as arbitrary.

In whatever manner the heat was at first distributed, thesystem of temperatures altering more and more, tends to coincidesensibly with a definite state which depends only on the form ofthe solid. In the ultimate state the temperatures of all the pointsare lowered in the same time, but preserve amongst each other thesame ratios: in order to express this property the analytical for-mulse contain terms composed of exponentials and of quantitiesanalogous to trigonometric functions.

Several problems of mechanics present analogous results, such asthe isochronism of oscillations, the multiple resonance of sonorousbodies. Common experiments had made these results remarked,and analysis afterwards demonstrated their true cause. As tothose results which depend on changes of temperature, they couldnot have been recognised except by very exact experiments ; butmathematical analysis has outrun observation, it has supplementedour senses, and has made us in a manner witnesses of regular andharmonic vibrations in the interior of bodies.

20. These considerations present a singular example of therelations which exist between the abstract science of numbersand natural causes.

When a metal bar is exposed at one end to the constant actionof a source of heat, and every point of it has attained its highesttemperature, the system of fixed temperatures corresponds exactlyto a table of logarithms; the numbers are the elevations of ther-mometers placed at the different points, and the logarithms arethe distances of these points from the source. In general heatdistributes itself in the interior of solids according to a simple lawexpressed by a partial differential equation common to physicalproblems of different order. The irradiation of heat has an evidentrelation to the tables of sines, for the rays which depart from thesame point of a heated surface, differ very much from each other,

- I.] INTRODUCTION. 23

and their intensity is rigorously proportional to the sine of theangle which the direction of each ray makes with the element ofsurface.

If we could observe the changes of temperature for every in-stant at every point of a solid homogeneous mass, we should dis-cover in these series of observations the properties of recurringseries, as of sines and logarithms; they would be noticed forexample in the diurnal or annual variations of temperature ofdifferent points of the earth near its surface.

We should recognise again the same results and all the chiefelements of general analysis in the vibrations of elastic media, inthe properties of lines or of curved surfaces, in the movements ofthe stars, and those of light or of fluids. Thus the functions ob-tained by successive differentiations, which are employed in thedevelopment of infinite series and in the solution of numericalequations, correspond also to physical properties. The first ofthese functions, or the fluxion properly so called, expresses ingeometry the inclination of the tangent of a curved line, and indynamics the velocity of a moving body when the motion varies ;in the theory of heat it measures the quantity of heat which flowsat each point of a body across a given surface. Mathematicalanalysis has therefore necessary relations with sensible phenomena;its object is not created by human intelligence; it is a pre-existentelement of the universal order, and is not in any way contingentor fortuitous; it is imprinted throughout all nature.

21. Observations more exact and more varied will presentlyascertain whether the effects of heat are modified by causes whichhave not yet been perceived, and the theory will acquire freshperfection by the continued comparison of its results with theresults of experiment; it will explain some important phenomenawhich we have not yet been able to submit to calculation; it willshew how to determine all the thermometric effects of the solarrays, the fixed or variable temperature which would be observed atdifferent distances from the equator, whether in the interior ofthe earth- or beyond the limits of the atmosphere, whether in theocean or in different regions of the air. From it will be derivedthe mathematical knowledge of the great movements which resultfrom the influence of heat combined with that of gravity. The

26 THEOHY OF HEAT. [CHAP. I.

same principles will serve to measure the conducibilities, proper orrelative, of different bodies, and their specific capacities, to dis-tinguish all the causes which modify the emission of heat at thesurface of solids, and to perfect thermometric instruments.

The theory of heat will always attract the attention of ma-thematicians, by the rigorous exactness of its elements and theanalytical difficulties peculiar to it, and above all by the extentand usefulness of its applications ; for all its consequences con-cern at the same time general physics, the operations of the arts,domestic uses and civil economy.

SECTION II.

Preliminary definitions and general notions.

22. O F the nature of heat uncertain hypotheses only could beformed, but the knowledge of the mathematical laws to which itseffects are subject is independent of all hypothesis; it requires onlyan attentive examination of the chief facts which common obser-vations have indicated, and which have been confirmed by exactexperiments.

It is necessary then to set forth, in the first place, the generalresults of observation, to give exact definitions of all the elementsof the analysis, and to establish the principles upon which thisanalysis ought to be founded.

The action of heat tends to expand all bodies, solid, liquid orgaseous; this is the property which gives evidence of its presence.Solids and liquids increase in volume, if the quantity of heat whichthey contain increases; they contract if it diminishes.

When all the parts of a solid homogeneous body, for examplethose of a mass of metal, are equally heated, and preserve withoutany change the same quantity of heat, they have also and retainthe same density. This state is expressed by saying that through-out the whole extent of the mass the molecules have a commonand permanent temperature.

23. The thermometer is a body whose smallest changes ofvolume can be appreciated ; it serves to measure temperatures by

SECT. II.] PRELIMINARY DEFINITIONS. 27

the dilatation of a fluid or of air. We assume the construction,use and properties of this instrument to be accurately known.The temperature of a body equally heated in every part, andwhich keeps its heat, is that which the thermometer indicateswhen it is and remains in perfect contact with the body inquestion.

Perfect contact is when the thermometer is completely im-mersed in a fluid mass, and, in general, when there is no point ofthe external surface of the instrument which is not touched by oneof the points of the solid or liquid mass whose temperature is to bemeasured. In experiments it is not always necessary that this con-dition should be rigorously observed; but it ought to be assumedin order to make the definition exact.

24. Two fixed temperatures are determined on, namely: thetemperature of melting ice which is denoted by 0? and the tem-perature of boiling water which we will denote by 1: the water issupposed to be boiling under an atmospheric pressure representedby a certain height of the barometer (76 centimetres), the mercuryof the barometer being at the temperature 0.

25. Different quantities of heat are measured by determininghow many times they contain a fixed quantity which is taken asthe unit. Suppose a mass of ice having a definite weight (a kilo-gramme) to be at temperature 0, and to be converted into water atthe same temperature 0 by the addition of a certain quantity ofheat: the quantity of heat thus added is taken as the unit ofmeasure. Hence the quantity of heat expressed by a number 0contains C times the quantity required to dissolve a kilogrammeof ice at the temperature zero into a mass of water at the samezero temperature.

26. To raise a metallic mass having a certain weight, a kilo-gramme of iron for example, from the temperature 0 to thetemperature 1, a new quantity of heat must be added to thatwhich is already contained in the mass. The number C whichdenotes this additional quantity of heat, is the specific capacity ofiron for heat; the number G has very different values for differentsubstances.


27. If a body of definite nature and weight (a kilogramme ofmercury) occupies a volume F a t temperature 0, it will oecupy agreater volume F + A, when it has acquired the temperature 1,that is to say, when the heat which it contained at the tempera-ture 0 has been increased by a new quantity C, equal to thespecific capacity of the body for heat. But if, instead of addingthis quantity (7, a quantity zG is' added (z being a numberpositive or negative) the new volume will be F-h S insteadof F + A. Now experiments shew that if z is equal to \y theincrease of volume 8 is only half the total increment A, andthat in general the value of 8 is z \ when the quantity of heatadded is zG.

28. The ratio z of the two quantities zG and G of heat added,which is the same as the ratio of the two increments of volume 8and A, is that which is called the temperature; hence the quantitywhich expresses the actual temperature of a body represents theexcess of its actual volume over the volume which it would occupyat the temperature of melting ice, unity representing the wholeexcess of volume which corresponds to the boiling point ofwater, over the volume which corresponds to the melting pointof ice.

29. The increments of volume of bodies are in general pro-portional to the increments of the quantities of heat whichproduce the dilatations, but it must be remarked that this propor-tion is exact only in the case where the bodies in question aresubjected to temperatures remote from those which determinetheir change of state. The application of these results to allliquids must not be relied on; and with respect to water inparticular, dilatations do not always follow augmentations ofheat.

In general the temperatures are numbers proportional to thequantities of heat added, and in the cases considered by us,these numbers are proportional also to the increments ofvolume.

30. Suppose that a body bounded by a plane surface havinga certain area (a square metre) is maintained in any manner


whatever at constant temperature 1, common to all its points,and that the surface in question is in contact with air maintainedat temperature 0 : the heat which escapes continuously at thesurface and passes into the surrounding medium will be replacedalways by the heat which proceeds from the constant cause towhose action the body is exposed; thus, a certain quantity of heatdenoted by h will flow through the surface in a definite time (aminute).

This amount h, of a flow continuous and always similar toitself, which takes place at a unit of surface at a fixed temperature,is the measure of the external conducibility of the body, that isto say, of the facility with which its surface transmits heat to theatmospheric air.

The air is supposed to be continually displaced with a givenuniform velocity: but if the velocity of the current increased, thequantity of heat communicated to the medium would vary also :the same would happen if the density of the medium wereincreased.

31. If the excess of the constant temperature of the bodyover the temperature of surrounding bodies, instead of being equalto 1, as has been supposed, had a less value, the quantity of heatdissipated would be less than h. The result of observation is,as we shall see presently, that this quantity of heat lost may beregarded as sensibly proportional to the excess of the temperatureof the body over that of the air and surrounding bodies. Hencethe quantity h having been determined by one experiment inwhich the surface heated is at temperature 1, and the medium attemperature 0; we conclude that hz would be the quantity, if thetemperature of the surface were z, all the other circumstancesremaining the same. This result must be admitted when z is asmall fraction.

32. The value h of the quantity of heat which is dispersedacross a heated surface is different for different bodies; and itvaries for the same body according to the different states of thesurface. The effect of irradiation diminishes as the surfacebecomes more polished; so that by destroying the polish of thesurface the value of h is considerably increased. A heated


metallic body will be more quickly cooled if its external surface iscovered with a black coating such as will entirely tarnish itsmetallic lustre.

33. The rays of heat which escape from the surface of a bodypass freely through spaces void of air; they are propagated alsoin atmospheric air: their directions are not disturbed by agitationsin the intervening air: they can be reflected by metal mirrorsand collected at their foci. Bodies at a high temperature, whenplunged into a liquid, heat directly only those parts of the masswith which their surface is in contact. The molecules whose dis-tance from this surface is not extremely small, receive no directheat; it is not the same with aeriform fluids; in these the rays ofheat are borne with extreme rapidity to considerable distances,whether it be that part of these rays traverses freely the layers ofair, or whether these layers transmit the rays suddenly withoutaltering their direction.

84. When the heated body is placed in air which is main-tained at a sensibly constant temperature, the heat communicatedto the air makes the layer of the fluid nearest to the surface of thebody lighter; this layer rises more quickly the more intensely it isheated, and is replaced by another mass of cool air. A currentis thus established in the air whose direction is vertical, andwhose velocity is greater as the temperature of the body is higher.For this reason if the body cooled itself gradually the velocity ofthe current would diminish with the temperature, and the lawof cooling would not be exactly the same as if the body wereexposed to a current of air at a constant velocity.

35. When bodies are sufficiently heated to diffuse a vivid light,part of their radiant heat mixed with that light can traverse trans-parent solids or liquids, and is subject to the force which producesrefraction. The quantity of heat which possesses this facultybecomes less as the bodies are less inflamed; it is, we may say,insensiblefor very opaque bodies however highly theymaybe heated.A thin transparent plate intercepts almost all the direct heatwhich proceeds from an ardent mass of metal; but it becomeshcnted in proportion as the intercepted rays are accumulated in


i t ; whence, if it is formed of ice, it becomes liquid; but if thisplate of ice is exposed to the rays of a torch it allows a sensibleamount of heat to pass through with the light.

36. We have taken as the measure of the external conduci-bility of a solid body a coefficient h, which denotes the quantity ofheat which would pass, in a definite time (a minute), from thesurface of this body, into atmospheric air, supposing that the sur-face had a definite extent (a square metre), that the constanttemperature of the body was 1, and that of the air 0, and thatthe heated surface was exposed to a current of air of a given in-variable velocity. This value of h is determined by observation.The quantity of heat expressed by the coefficient is composed oftwo distinct parts which cannot be measured except by very exactexperiments. One is the heat communicated by way of contact tothe surrounding air: the other, much less than the first, is theradiant heat emitted. We must assume, in our first investigations,that the quantity of heat lost does not change when the tempera-tures of the body and of the medium are augmented by the samesufficiently small quantity.

37. Solid substances differ again, as we have already remarked,by their property of being more or less permeable to heat; thisquality is their conducibility proper: we shall give its definition andexact measure, after having treated of the uniform and linear pro-pagation of heat. Liquid substances possess also the property oftransmitting heat from molecule to molecule, and the numericalvalue of their conducibility varies according to the nature of thesubstances : but this effect is observed with difficulty in liquids,since their molecules change places on change of temperature. Thepropagation of heat in them depends chiefly on this continual dis-placement, in all cases where the lower parts of the mass are mostexposed to the action of the source of heat. If, on the contrary,the source of heat be applied to that part of the mass which ishighest, as was the case in several of our experiments, the transferof heat, which is very slow, does not produce any displacement,at least when the increase of temperature does not diminish thevolume, as is indeed noticed in singular cases bordering on changesof state.


38. To this explanation of the chief results of observation, ageneral remark must be added on equilibrium of temperatures;which consists in this, that different bodies placed in the same re-gion, all of whose parts are and remain equally heated, acquire alsoa common and permanent temperature.

Suppose that all the parts of a mass M have a common andconstant temperature a, which is maintained by any cause what-ever : if a smaller body m be placed in perfect contact with themass My it will assume the common temperature a.

In reality this result would not strictly occur except after aninfinite time : but the exact meaning of the proposition is that ifthe body m had the temperature a before being placed in contact,it would keep it without any change. The same would be the casewith a multitude of other bodies n, p, q, r each of which wasplaced separately in perfect contact with the mass M: all wouldacquire the constant temperature a. Thus a thermometer if suc-cessively applied to the different bodies m, n,p, q, r would indicatethe same temperature.

39. The effect in question is independent of contact, andwould still occur, if every part of the body m were enclosed inthe solid M, as in an enclosure, without touching any of its parts.For example, if the solid were a spherical envelope of a certainthickness, maintained by some external cause at a temperature a,and containing a space entirely deprived of air, and if the body mcould be placed'in any part whatever of this spherical space, with-out touching any point of the internal surface of the enclosure, itwould acquire the common temperature a, or rather, it would pre-serve it if it had it already. The result would be the same forall the other bodies n, py q, r, whether they were placed separatelyor all together in the same enclosure, and whatever also their sub-stance and form might be.

40. Of all modes of presenting to ourselves the action ofheat, that which seems simplest and most conformable to observa-tion, consists in comparing this action to that of light. Mole-cules separated from one another reciprocally communicate, acrossempty space, their rays of heat, just as shining bodies transmittheir light.

SECT. II.] GENERAL NOTIONS. 33

If within an enclosure closed in all directions, and maintainedby some external cause at a fixed temperature a, we suppose dif-ferent bodies to be placed without touching any part of the bound-ary, different effects will be observed according as the bodies,introduced into this space free from air, are more or less heated.If, in the first instance, we insert only one of these bodies, at thesame temperature as the enclosure, it will send from all points ofits surface as much heat as it receives from the solid which sur-rounds it, and is maintained in its original state by this exchangeof equal quantities.

If we insert a second body whose temperature b is less than a,it will at first receive from the surfaces which surround it onall sides without touching it, a quantity of heat greater than thatwhich it gives out: it will be heated more and more and willabsorb through its surface more heat than in the first instance.

The initial temperature b continually rising, will approach with-out ceasing the fixed temperature a, so that after a certain timethe difference will be almost insensible. The effect would be op-posite if we placed within the same enclosure a third body whosetemperature was greater than a.

41. All bodies have the property of emitting heat throughtheir surface; the hotter they are the more they emit; theintensity of the emitted rays changes very considerably with thestate of the surface.

42. Every surface which receives rays of heat from surround-ing bodies reflects part and admits the rest: the heat which is notreflected, but introduced through the surface, accumulates withinthe solid; and so long as it exceeds the quantity dissipated byirradiation, the temperature rises.

43. The rays which tend to go out of heated bodies arearrested at the surface by a force which reflects part of them intothe interior of the mass. The cause which hinders the incidentrays from traversing the surface, and which divides these rays intotwo parts, of which one is reflected and the other admitted, acts inthe same manner on the rays which are directed from the interiorof the body towards external space.

F. H. 3


If by modifying the state of the surface we increase the forceby which it reflects the incident rays, we increase at the same timethe power which it has of reflecting towards the interior of thebody rays which are tending to go out. The incident rays intro-duced into the mass, and the rays emitted through the surface, areequally diminished in quantity.

44. If within the enclosure above mentioned a number ofbodies were placed at the same time, separate from each otherand unequally heated, they would receive and transmit rays of heatso that at each exchange their temperatures would continuallyvary, and would all tend to become equal to the fixed temperatureof the enclosure.

This effect is precisely the same as that which occurs whenheat is propagated within solid bodies; for the molecules whichcompose these bodies are separated by spaces void of air, andhave the property of receiving, accumulating and emitting heat.Each of them sends out rays on all sides, and at the same timereceives other rays from the molecules which surround it.

45. The heat given out by a point situated in the interior ofa solid mass can pass directly to an extremely small distance only;it is, we may say, intercepted by the nearest particles; these parti-cles only receive the heat directly and act on more distant points.It is different with gaseous fluids; the direct effects of radiationbecome sensible in them at very considerable distances.

46. Thus the heat which escapes in all directions from a partof the surface of a solid, passes on in air to very distant points; butis emitted only by those molecules of the body which are extremelynear the surface. A point of a heated mass situated at a verysmall distance from the plane superficies which separates the massfrom external space, sends to that space an infinity of rays, butthey do not all arrive there; they are diminished by all that quan-tity of heat which is arrested by the intermediate molecules of thesolid. The part of the ray actually dispersed into space becomesless according as it traverses a longer path within the mass. Thusthe ray which escapes perpendicular to the surface has greater in-tensity than that which, departing from the same point, follows


an oblique direction, and the most oblique rays are wholly inter-cepted.

The same consequences apply to all the points which are nearenough to the surface to take part in the emission of heat, fromwhich it necessarily follows that the whole quantity of heat whichescapes from the surface in the normal direction is very muchgreater than that whose direction is oblique. We have submittedthis question to calculation, and our analysis proves that the in-tensity of the ray is proportional to the sine of the angle whichthe ray makes with the element of surface. Experiments hadalready indicated a similar result.

47. This theorem expresses a general law which has a neces-sary connection with the equilibrium and mode of action of heat.If the rays which escape from a heated surface had the same in-tensity in all directions, a thermometer placed at one of the pointsof a space bounded on all sides by an enclosure maintained at aconstant temperature would indicate a temperature incomparablygreater than that of the enclosure1. Bodies placed within thisenclosure would not take a common temperature, as is alwaysnoticed; the temperature acquired by them would depend on theplace which they occupied, or on their form, or on the forms ofneighbouring bodies.

The same results would be observed, or other effects equallyopposed to common experience, if between the rays which escapefrom the same point any other relations were admitted differentfrom those which we have enunciated. We have recognised thislaw as the only one compatible with the general fact of the equi-librium of radiant heat.

48. If a space free from air is bounded on all sides by a solidenclosure whose parts are maintained at a common and constanttemperature a, and if a thermometer, having the actual tempera-ture a, is placed at any point whatever of the space, its temperaturewill continue without any change. It will receive therefore ateach instant from the inner surface of the enclosure as much heatas it gives out to it. This effect of the rays of heat in a givenspace is, properly speaking, the measure of the temperature: but

i See proof by M. Fourier, Ann. d. Ch. et Ph. Ser. 2, iv. p. 128. [A. F.]g 9


this consideration presupposes the mathematical theory of radiantheat.

If now between the thermometer and a part of the surface ofthe enclosure a body M be placed whose temperature is a, thethermometer will cease to receive rays from one part of the innersurface, but the rays will be replaced by those which it will re-ceive from the interposed body M. An easy calculation provesthat the compensation is exact, so that the state of the thermo-meter will be unchanged. It is not the same if the temperatureof the body M is different from that of the enclosure. Whenit is greater, the rays which the interposed body M sends to thethermometer and which replace the intercepted rays convey moreheat than the latter; the temperature of the thermometer musttherefore rise.

If, on the contrary, the intervening body has a temperatureless than a, that of the thermometer must fall; for the rays whichthis body intercepts are replaced by those which it gives out, thatis to say, by rays cooler than those of the enclosure; thus thethermometer does not receive all the heat necessary to maintainits temperature a.

49. Up to this point abstraction has been made of the powerwhich all surfaces have of reflecting part of the rays which aresent to them. If this property were disregarded we should haveonly a very incomplete idea of the equilibrium of radiant heat.

Suppose then that on the inner surface of the enclosure, main-tained at a constant temperature, there is a portion which enjoys,in a certain degree, the power in question; each point of the re-flecting surface will send into space two kinds of rays; the one goout from the very interior of the substance of which the enclosure isformed, the others are merely reflected by the same surface againstwhich they had been sent. But at the same time that the surfacerepels on the outside part of the incident rays, it retains in themside part of its own rays. In this respect an exact compensationis established, that is to say, every one of its own rays which thesurface hinders from going out is replaced by a reflected ray ofequal intensity.

The same result would happen, if the power of reflecting raysaffected in any degree whatever other parts of the enclosure, or the


surface of bodies placed within the same space and already atthe common temperature.

Thus the reflection of heat does not disturb the equilibriumof temperatures, and does not introduce, whilst that equilibriumexists, any change in the law according to which the intensity ofrays which leave the same point decreases proportionally to thesine of the angle of emission.

50. Suppose that in the same enclosure, all of whose partsmaintain the temperature a, we place an isolated body M, anda polished metal surface Ry which, turning its concavity towardsthe body, reflects great part of the rays which it received from thebody; if we place a thermometer between the body Mand the re-flecting surface R, at the focus of this mirror, three different effectswill be observed according as the temperature of the body M isequal to the common temperature a, or is greater or less.

In the first ease, the thermometer preserves the temperaturea; it receives 1°, rays of heat from all parts of the enclosure nothidden from it by the body M or by the mirror; 2°, rays given outby the body; 3°, those which the surface R sends out to the focus,whether they come from the mass of the mirror itself, or whether itssurface has simply reflected them; and amongst the last we maydistinguish between those which have been sent to the mirror bythe mass M, and those which it has received from the enclosure.All the rays in question proceed from surfaces which, by hypo-thesis, have a common temperature a, so that the thermometeris precisely in the same state as if the space bounded by the en-closure contained no other body but itself.

In the second case, the thermometer placed between the heatedbody M and the mirror, must acquire a temperature greater thana. In reality, it receives the same rays as in the first .hypothesis;but with two remarkable differences: one arises from the fact thatthe rays sent by the body M to the mirror, and reflected upon thethermometer, contain more heat than in the first case. The otherdifference depends on the fact that the rays sent directly by thebody M to the thermometer contain more heat than formerly.Both causes, and chiefly the first, assist in raising the tempera-ture of the thermometer.

In the third case, that is to say, when the temperature of the


mass M is less than a, the temperature must assume also a tem-perature less than a. In fact, it receives again all the varieties ofrays which we distinguished in the first case : but there are twokinds of them which contain less heat than in this first hypothesis,that is to say, those which, being sent out by the body My arereflected by the mirror upon the thermometer, and those whichthe same body M sends to it directly. Thus the thermometer doesnot receive all the heat which it requires to preserve its originaltemperature a. I t gives out more heat than it receives. It isinevitable then that its temperature must fall to the point atwhich the rays which it receives suffice to compensate those whichit loses. This last effect is what is called the reflection of cold,and which, properly speaking, consists in the reflection of toofeeble heat. The mirror intercepts a certain quantity of heat, and-replaces it by a less quantity.

51. If in the enclosure, maintained at a constant temperaturea, a body M be placed, whose temperature a is less than a, thepresence of this body will lower the thermometer exposed to itsrays, and we may remark that the rays sent to the thermometerfrom the surface of the body My are in general of two kinds,namely, those which come from inside the mass M3 and thosewhich, coming from different parts of the enclosure, meet the sur-face M and are reflected upon the thermometer. The latter rayshave the common temperature a, but those which belong to thebody M contain less heat, and these are the rays which cool thethermometer. If now, by changing the state of the surface of thebody M, for example, by destroying the polish, we diminish thepower which it has of reflecting the incident rays, the thermo-meter will fall still lower, and will assume a temperature a" lessthan a. In fact all the conditions would be the same as in thepreceding case, if it were not that the body M gives out a greaterquantity of its own rays and reflects a less quantity of the rayswhich it receives from the enclosure; that is to say, these last rays,which have the common temperature, are in part replaced bycooler rays. Hence the thermometer no longer receives so muchheat as formerly.

If, independently of the change in the surface of the body M,we place a metal mirror adapted to reflect upon the thermometer


the rays which have left M, the temperature will assume a valuea" less than a". The mirror, in fact, intercepts from the thermo-meter part of the rays of the enclosure which all have the tem-perature a, and replaces them by three kinds of rays ; namely,1°, those which come from the interior of the mirror itself, andwhich have the common temperature ; 2°, those which the differentparts of the enclosure send to the mirror with the same tempera-ture, and which are reflected to the focus ; 3°, those which, comingfrom the interior of the body M, fall upon the mirror, and arereflected upon the thermometer. The last rays have a tempera-ture less than a ; hence the thermometer no longer receives somuch heat as it received before the mirror was set up.

Lastly, if we proceed to change also the state of the surface ofthe mirror, and by giving it a more perfect polish, increase itspower of reflecting heat, the thermometer will fall still lower. Infact, all the conditions exist which occurred in the preceding case.Only, it happens that the mirror gives out a less quantity of itsown rays, and replaces them by those which it reflects. Now,amongst these last rays, all those which proceed from the interiorof the mass M are less intense than if they had come from theinterior of the metal mirror; hence the thermometer receives stillless heat than formerly: it will assume therefore a temperaturea!'" less than d\

By the same principles all the known facts of the radiation ofheat or of cold are easily explained.

52. The effects of heat can by no means be compared withthose of an elastic fluid whose molecules are at rest.

It would be useless to attempt to deduce from this hypothesisthe laws of propagation which we have explained in this work,and which all experience has confirmed. The free state of heat isthe same as that of light; the active state of this element is thenentirely different from that of gaseous substances. Heat acts inthe same manner in a vacuum, in elastic fluids, and in liquid orsolid masses, it is propagated only by way of radiation, but itssensible effects differ according to the nature of bodies.

53. Heat is the origin of all elasticity; it is the repulsiveforce which preserves the form of solid masses, and the volume of


liquids. In solid masses, neighbouring molecules would yield totheir mutual attraction, if its effect were not destroyed by theheat which separates them.

This elastic force is greater according as the temperature ishigher; which is the reason why bodies dilate or contract whentheir temperature is raised or lowered.

54 The equilibrium which exists, in the interior of a solidmass, between the repulsive force of heat and the molecular attrac-tion, is stable ; that is to say, it re-establishes itself when disturbedby an accidental cause. If the molecules are arranged at distancesproper for equilibrium, and if an external force begins to increasethis distance without any change of temperature, the effect ofattraction begins by surpassing that of heat, and brings back themolecules to their original position, after a multitude of oscillationswhich become less and less sensible.

A similar effect is exerted in the opposite sense when a me-chanical cause diminishes the primitive distance of the molecules ;such is the origin of the vibrations of sonorous or flexible bodies,and of all the effects of their elasticity.

55. In the liquid or gaseous state of matter, the externalpressure is additional or supplementary to the molecular attrac-tion, and, acting on the surface, does not oppose change of form,but only change of the volume occupied. Analytical investigationwill best shew how the repulsive force of heat, opposed to theattraction of the molecules or to the external pressure, assists inthe composition of bodies, solid or liquid, formed of one or moreelements, and determines the elastic properties of gaseous fluids;but these researches do not belong to the object before us, andappear in dynamic theories.

56. I t cannot be doubted that the mode of action of heatalways consists, like that of light, in the reciprocal communicationof rays, and this explanation is at the present time adopted bythe majority of physicists; but it is not necessary to consider thephenomena under this aspect in order to establish the theory of heat.In the course of this work it will be seen how the laws of equili-brium and propagation of radiant heat, in solid or liquid masses,

SECT. III.] PRINCIPLE OF COMMUNICATION. 41

can be rigorously demonstrated, independently of any physicalexplanation, as the necessary consequences of common observations.

SECTION III.

Principle of the communication of heat.

57. We now proceed to examine what experiments teach, usconcerning the communication of heat.

If two equal molecules are formed of the same substance andhave the same temperature, each of them receives from the otheras much heat as it gives up to i t ; their mutual action may then beregarded as null, since the result of this action can bring about nochange in the state of the molecules. If, on the contrary, the firstis hotter than the second, it sends to it more heat than it receivesfrom it; the result of the mutual action is the difference of thesetwo quantities of heat. In all cases we make abstraction ofthe two equal quantities of heat which any two material pointsreciprocally give up ; we conceive that the point most heatedacts only on the other, and that, in virtue of this action, the firstloses a certain quantity of heat which is acquired by the second.Thus the action of two molecules, or the quantity of heat whichthe hottest communicates to the other, is the difference of the twoquantities which they give up to each other.

58. Suppose that we place in air a solid homogeneous body,whose different points have unequal actual temperatures; each ofthe molecules of which the body is composed will begin to receiveheat from those which are at extremely small distances, or willcommunicate it to them. This action exerted during the sameinstant between all points of the mass, will produce an infinitesi-mal resultant change in all the temperatures: the solid will ex-perience at each instant similar effects, so that the variations oftemperature will become more and more sensible.

Consider only the system of two molecules, m and n, equal andextremely near, and let us ascertain what quantity of heat thefirst can receive from the second during one instant: we maythen apply the same reasoning to all the other points which are

near enough to the point m, to act directly on it during the firstinstant.

The quantity of heat communicated by the point n to thepoint m depends on the duration of the instant, on the very smalldistance between these points, on the actual temperature of eachpoint, and on the nature of the solid substance ; that is to say, ifone of these elements happened to vary, all the other remainingthe same, the quantity of heat transmitted would vary also. Nowexperiments have disclosed, in this respect, a general result: itconsists in this, that all the other circumstances being the same,the quantity of heat which one of the molecules receives from theother is proportional to the difference of temperature of the twomolecules. Thus the quantity would be double, triple, quadruple, ifeverything else remaining the same, the difference of the tempera-ture of the point n from that of the point m became double, triple,or quadruple. To account for this result, we must consider that theaction of n on m is always just as much greater as there is a greaterdifference between the temperatures of the two points: it is null,if the temperatures are equal, but if the molecule n contains moreheat than the equal molecule m, that is to say, if the temperatureof m being v} that of n is v + A, a portion of the exceeding heatwill pass from n to m. Now, if the excess of heat were double, or,which is the same thing, if the temperature of n were v + 2 A, theexceeding heat would be composed of two equal parts correspond-ing to the two halves of the whole difference of temperature 2A;each of these parts would have its proper effect as if it aloneexisted : thu& the quantity of heat communicated by n to m wouldbe twice as great as when the difference of temperature is only A.This simultaneous action of the different parts of the exceedingheat is that which constitutes the principle of the communicationof heat. It follows from it that the sum of the partial actions, orthe total quantity of heat which m receives from n is proportionalto the difference of the two temperatures.

59. Denoting by v and v the temperatures of two equal mole-cules m and n, by p, their extremely small distance, and by dt, theinfinitely small duration of the instant, the quantity of heat whichm receives from n during this instant will be expressed by(v -v) <j> (p) .dt. We denote by </> (p) a certain function of the

SECT. III.] PRINCIPLE OF COMMUNICATION. 43

distance p which, in solid bodies and in liquids, becomes nothingwhen p has a sensible magnitude. The function is the same forevery point of the same given substance; it varies with the natureof the substance.

60. The quantity of heat which bodies lose through their sur-face is subject to the same principle. If we denote by <r the area,finite or infinitely small, of the surface, all of whose points havethe temperature v, and if a represents the temperature of theatmospheric air, the coefficient h being the measure of the ex-ternal conducibility, we shall have ah (v — a) dt as the expressionfor the quantity of heat which this surface a transmits to the airduring the instant dt.

When the two molecules, one of which transmits to the othera certain quantity of heat, belong to the same solid, the exactexpression for the heat communicated is that which we havegiven in the preceding article; and since the molecules areextremely near, the difference of the temperatures is extremelysmall. It is not the same when heat passes from a solid body intoa gaseous medium. But the experiments teach us that if thedifference is a quantity sufficiently small, the heat transmitted issensibly proportional to that difference, and that the number hmay, in these first researches *, be considered as having a constantvalue, proper to each state of the surface, but independent of thetemperature.

61. These propositions relative to the quantity of heat com-municated have been derived from different observations. Wesee first, as an evident consequence of the expressions in question,that if we increased by a common quantity all the initial tempe-ratures of the solid mass, and that of the medium in which it isplaced, the successive changes of temperature would be exactlythe same as if this increase had not been made. Now this resultis sensibly in accordance with experiment; it has been admittedby the physicists who first have observed the effects of heat. .

1 More exact laws of cooling investigated experimentally by Dulong and Petitwill be found in the Journal de VEcole Poly technique, Tome xi. pp. 234—294,Paris, 1820, or in Jamin, Cours de Physique, Legon 47. [A. F.]


62. If the medium is maintained at a constant temperature,and if the heated body which is placed in that medium hasdimensions sufficiently small for the temperature, whilst fallingmore and more, to remain sensibly the same at all points of thebody, it follows from the same propositions, that a quantity of heatwill escape at each instant through the surface of the body pro-portional to the excess of its actual temperature over that of themedium. Whence it is easy to conclude, as will be seen in thecourse of this work, that the line whose abscissae represent thetimes elapsed, and whose ordinates represent the temperaturescorresponding to those times, is a logarithmic curve: now, ob-servations also furnish the same result, when the excess of thetemperature of the solid over that of the medium is a sufficientlysmall quantity.

63. Suppose the medium to be maintained at the constanttemperature 0, and that the initial temperatures of differentpoints a, b, c, d &c. of the same mass are a, /3, y, 8 &c, that at theend of the first instant they have become a, ft', y, S' &c, that atthe end of the second instant they have become a", /3'r, y'} 8" &c,and so on. We may easily conclude from the propositions enun-ciated, that if the initial temperatures of the same points hadbeen g%, g/3, gy, g8 &c. (g being any number whatever), theywould have become, at the end of the first instant, by virtue ofthe action of the different points, gz', g/3', 97', g$ &c, and at theend of the second instant, go.", g/3", gy", gS" &c, and so on. Forinstance, let us compare the case when the initial temperaturesof the points, a, b, c> d &c. were a, /3, 7, S &c. with that in whichthey are 2a, 2/3, 27, 2S &c, the medium preserving in both casesthe temperature 0. In the second hypothesis, the difference ofthe temperatures of any two points whatever is double what itwas in the first, and the excess of the temperature of each point,over that of each molecule of the medium, is also double; con-sequently the quantity of heat which any molecule whateversends to any other, or that which it receives, is, in the secondhypothesis, double of that which it was in the first. The changeof temperature which each point suffers being proportional to thequantity of heat acquired, it follows that, in the second case, thischange is double what it was in the first case. Now we have

SECT. IV.J UNIFORM LINEAR MOVEMENT. 45

supposed that the initial temperature of the first point, which wasa, became a! at the end of the first instant; hence if this initialtemperature had been 2a, and if all the other temperatures hadbeen doubled, it would have become 2 a'. The same would be thecase with all the other molecules b, c, d, and a similar resultwould be derived, if the ratio instead of being 2, were any numberwhatever g. I t follows then, from the principle of the communica-tion of heat, that if we increase or diminish in any given ratioall the initial temperatures, we increase or diminish in the sameratio all the successive temperatures.

This, like the two preceding results, is confirmed by observa-tion. I t could not have existed if the quantity of heat whichpasses from one molecule to another had not been, actually, pro-portional to the difference of the temperatures.

64. Observations have been made with accurate instruments,on the permanent temperatures at different points of a bar or of ametallic ring, and on the propagation of heat in the same bodiesand in several other solids of the form of spheres or cubes. Theresults of these experiments agree with those which are derivedfrom the preceding propositions. They would be entirely differ-ent if the quantity of heat transmitted from one solid molecule toanother, or to a molecule of air, were not proportional to theexcess of temperature. It is necessary first to know all therigorous consequences of this proposition; by it we determine thechief part of the quantities which are the object of the problem.By comparing then the calculated values with those given bynumerous and very exact experiments, we can easily measure thevariations of the coefficients, and perfect our first researches.

SECTION IV.

On the uniform and linear movement of heat

65. We shall consider, in the first place, the uniform move-ment of heat in the simplest case, which is that of an infinitesolid enclosed between two parallel planes.

We suppose a solid body formed of some homogeneous sub-stance to be enclosed between two parallel and infinite planes;


the lower plane A is maintained, by any cause whatever, at aconstant temperature a; we may imagine for example that themass is prolonged, and that the plane A is a section common tothe solid and to the enclosed mass, and is heated at all its pointsby a constant source of heat; the upper plane B is also main-tained by a similar cause at a fixed temperature b, whose value isless than that of a; the problem is to determine what would bethe result of this hypothesis if it were continued for an infinitetime,

If we suppose the initial temperature of all parts of this bodyto be b, it is evident that the heat which leaves the source A willbe propagated farther and farther and will raise the temperatureof the molecules included between the two planes: but the tem-perature of the upper plane being unable, according to hypothesisto rise above 6, the heat will be dispersed within the cooler mass,contact with which keeps the plane B at the constant temperatureb. The system of temperatures will tend more and more to afinal state, which it will never attain, but which would have theproperty, as we shall proceed to shew, of existing and keepingitself up without any change if it were once formed.

In the final and fixed state, which we are considering, the per-manent temperature of a point of the solid is evidently the sameat all points of the same section parallel to the base; and weshall prove that this fixed temperature, common to all the pointsof an intermediate section, decreases in arithmetic progressionfrom the base to the upper plane, that is to say, if we representthe constant temperatures a and b by the ordinates AOL and Bj3

B J

Fig. 1.

(see Fig. 1), raised perpendicularly to the distance AB between thetwo planes, the fixed temperatures of the intermediate layers willbe represented by the ordinates of the straight line a/3 which

SECT. IV.] UNIFORM LINEAR MOVEMENT. 47

joins the extremities a and /3; thus, denoting by z the height ofan intermediate section or its perpendicular distance from theplane A, by e the whole height or distance ABy and by v thetemperature of the section whose height is z, we must have the

equation v = a-\ z.6

In fact, if the temperatures were at first established in accord-ance with this law, and if the extreme surfaces A and B werealways kept at the temperatures a and &, no change wouldhappen in the state of the solid. To convince ourselves of this,it will be sufficient to compare the quantity of heat which wouldtraverse an intermediate section A' with that which, during thesame time, would traverse another section B'.

Bearing in mind that the final state of the solid is formedand continues, we see that the part of the mass which is belowthe plane A' must communicate heat to the part which is abovethat plane, since this second part is cooler than the first.

Imagine two points of the solid, m and m, very near to eachother, and placed in any manner whatever, the one m below theplane A\ and the other m above this plane, to be exerting theiraction during an infinitely small instant: m the hottest pointwill communicate to m a certain quantity of heat which willcross the plane A\ Let x, y, z be the rectangular coordinatesof the point m, and x, y> z the coordinates of the point m :consider also two other points n and n very near to each other,and situated with respect to the plane B\ in the same mannerin which m and m are placed with respect to the plane A': thatis to say, denoting by f the perpendicular distance of the twosections A! and B', the coordinates of the point n will be x, y, z + £and those of the point ri9 x\ yy z + £ ; the two distances mmand nn will be equal: further, the difference of the temperaturev of the point m above the temperature vf of the point m willbe the same as the difference of temperature of the two pointsn and ri. In fact the former difference will be determined bysubstituting first z and then z in the general equation

b — a— z,

and subtracting the second equation from the first, whence the


result v - v =^=^-(z - z). We shall then find, by the sub-

stitution of z + £ and z + £ that the excess of temperature ofthe point n over that of the point n is also expressed by

b — a , ,x

It follows from this that the quantity of heat sent by thepoint m to the point m will be the same as the quantity of heatsent by the point n to the point n, for all the elements whichconcur in determining this quantity of transmitted heat are thesame.

It is manifest that we can apply the same reasoning to everysystem of two molecules which communicate heat to each otheracross the section A' or the section B'; whence, if we couldsum up the whole quantity of heat which flows, during the sameinstant, across the section Af or the section B\ we should findthis quantity to be the same for both sections.

From this it follows that the part of the solid included be-tween A and B receives always as much heat as it loses, andsince this result is applicable to any portion whatever of themass included between two parallel sections, it is evident thatno part of the solid can acquire a temperature higher than thatwhich it has at present. Thus, it has been rigorously demon-strated that the state of the prism will continue to exist just as itwas at first.

Hence, the permanent temperatures of different sections of asolid enclosed between two parallel infinite planes, are representedby the ordinates of a straight line a/3, and satisfy the linear

,. b — aequation v = a H z.

66. By what precedes we see distinctly what constitutesthe propagation of heat in a solid enclosed between two paralleland infinite planes, each of which is maintained at a constanttemperature. Heat penetrates the mass gradually across thelower plane: the temperatures of the intermediate sections areraised, but can never exceed nor even quite attain a certainlimit which they approach nearer and nearer: this limit or finaltemperature is different for different intermediate layers, and


decreases in arithmetic progression from the fixed temperatureof the lower plane to the fixed temperature of the upper plane.

The final temperatures are those which would have to begiven to the solid in order that its state might be permanent;the variable state which precedes it may also be submitted toanalysis, as we shall see presently: but we are now consideringonly the system of final and permanent temperatures. In thelast state, during each division of time, across a section parallelto the base, or a definite portion of that section, a certainquantity of heat flows, which is constant if the divisions of time.are equal. This uniform flow is the same for all the intermediatesections ; it is equal to that which proceeds from the source, andto that which is lost during the same time, at the upper surfaceof the solid, by virtue of the cause which keeps the temperatureconstant.

67. The problem now is to measure that quantity of heatwhich is propagated uniformly within the solid, during a giventime, across a definite part of a section parallel to the base : itdepends, as we shall see, on the two extreme temperatures aand b, and on the distance e between the two sides of the solid;it would vary if any one of these elements began to change, theother remaining the same. Suppose a second solid to be formedof the same substance as the first, and enclosed between two

Fig. 2.

infinite parallel planes, whose perpendicular distance is e (seefig. 2) : the lower side is maintained at a fixed temperature a,and the upper side at the fixed temperature V; both solids areconsidered to be in that final and permanent state which hasthe property of maintaining itself as soon as it has been formed.

F. H. *


Thus the law of the temperatures is expressed for the first body

by the equation v = a + — ^ z, and for the second, by the equa-

7 / /

tion u = a + —— z, v in the first solid, and u in the second, beinge

the temperature of the section whose height is z.This arranged, we will compare the quantity of heat which,

during the unit of time traverses a unit of area taken on'anintermediate section L of the first solid, with that which duringthe same time traverses an equal area taken on the section IIof the second, e being the height common to the two sections,that is to say, the distance of each of them from their ownbase. "We shall consider two very near points n and n in thefirst body, one of which n is below the plane L and the otherri above this plane : x, yy z are the co-ordinates of n : and x, y, zthe co-ordinates of n, e being less than z'f and greater than z.

We shall consider also in the second solid the instantaneousaction of two points p and p', which are situated, with respectto the section L\ in the same manner as the points w and w' withxespect to the section L of the first solid. Thus the same co-ordinates x, yy z, and x\ y\ z referred to three rectangular axesin the second body, will fix also the position of the points pand p\

Now, the distance from the point n to the point n' is equalto the distance from the point p to the point p', and since thetwo bodies are formed of the same substance, we conclude, ac-cording to the principle of the communication of heat, that theaction of n on n\ or the quantity of heat given by n to n\ andthe action of p on p, are to each other in the same ratio as thedifferences of the temperature v — v and u — u.

Substituting v and then v' in the equation which belongs to

the first solid, and subtracting, we find v — v=—— (s — z)\ we

have also by means of the second equation u — v! = ~~ta [z - z'),

whence the ratio of the two actions in question is that of a toe

/ 7 /

a —o


We may now imagine many other systems of two molecules,the first of which sends to the second across the plane L, a certainquantity of heat, and each of these systems, chosen in the firstsolid, may be compared with a homologous system situated in thesecond, and whose action is exerted across the section I!; wecan then apply again the previous reasoning to prove that the

ratio of the two actions is always that of a~~ to a ~~, .J e e

Now, the whole quantity of heat which, during one instant,crosses the section L, results from the simultaneous action of amultitude of systems each of which is formed of two points;hence this quantity of heat and that which, in the second solid,crosses during the same instant the section L\ are also to each

,, . ,, ,. £a-b a-Vother in the ratio ol to —-,— .

e eIt is easy then to compare with each other" the intensities of

the constant flows of heat which are propagated uniformly in thetwo solids, that is to say, the quantities of heat which, duringunit of time, cross unit of surface of each of these bodies. Theratio of these intensities is that of the two quotients and

,—. If the two quotients are equal, the flows are the same,

whatever in other respects the values a, 6, ey a, b', e, may be ;in general, denoting the first flow by F and the second by F',

, ,, , F a-b a'-Vwe shall have y=a = -; -,—.

k Q e

68. Suppose that in the second solid, the permanent tempera-ture a of the lower plane is that of boiling water, 1; that thetemperature e of the upper plane is that of melting ice, 0; thatthe distance e of the two planes is the unit of measure (ametre); let us denote by K the constant flow of heat which,during unit of time (a minute) would cross unit of surface inthis fast solid, if it were formed of a given substance; K ex-pressing a certain number of units of heat, that is to say a certainnumber of times the heat necessary to convert a kilogrammeof ice into water: we shall have, in general, to determine the

4—2


constant flow F, in a solid formed of the same substance, theF a-b -p ^a-h

equation -~= or J? = A ——• .

The value of F denotes the quantity of heat which, duringthe unit of time, passes across a unit of area of the surface takenon a section parallel to the base.

Thus the thermometric state of a solid enclosed between twoparallel infinite plane sides whose perpendicular distance is e>and which are maintained at fixed temperatures a and b, isrepresented by the two equations :

v = a + z, and F=K- or F=-K-T.e e dz

The first of these equations expresses the law according towhich the temperatures decrease from the lower side to theopposite side, the second indicates the quantity of heat which,during a given time, crosses a definite part of a section parallelto the base.

69. We have taken this coefficient K, which enters intothe second equation, to be the measure of the specific conduci-bility of each substance; this number has very different valuesfor different bodies.

It represents, in general, the quantity of heat which, in ahomogeneous solid formed of a given substance and enclosedbetween two infinite parallel planes, flows, during one minute,across a surface of one square metre taken on a section parallelto the extreme planes, supposing that these two planes are main-tained, one at the temperature of boiling water, the other atthe temperature of melting ice, and that all the intermediateplanes have acquired and retain a permanent temperature.

We might employ another definition of conducibility, sincewe could estimate the capacity for heat by referring it to unitof volume, instead of referring it to unit of mass. All thesedefinitions are equally good provided they are clear and pre-cise.

We shall shew presently how to determine by observation thevalue K of the conducibility or conductibility in different sub-stances.


70. In order to establish the equations which we havecited in Article 68, it would not be necessary to suppose thepoints which exert their action across the planes to be at ex-tremely small distances.

The results would still be the same if the distances of these,points had any magnitude whatever; they would therefore applyalso to the case where the direct action of heat extended withinthe interior of the mass to very considerable distances, all thecircumstances which constitute the hypothesis remaining in otherrespects the same.

We need only suppose that the cause which maintains thetemperatures at the surface of the solid, affects not only thatpart of the mass which is extremely near to the surface, but that

its action extends to a finite depth. The equation v = a— a~ z

will still represent in this case the permanent temperatures ofthe solid. The true sense of this proposition is that, if we giveto all points of the mass the temperatures expressed by theequation, and if besides any cause whatever, acting on the twoextreme laminae, retained always every one of their moleculesat the temperature which the same equation assigns to them,the interior points of the solid would preserve without any changetheir initial state.

If we supposed that the action of a point of the mass couldextend to a finite distance e, it would be necessary that thethickness of the extreme laminge, whose state is maintained bythe external cause, should be at least equal to 6. But thequantity e having in fact, in the natural state of solids, onlyan inappreciable value, we may make abstraction of this thick-ness; and it is sufficient for the external cause to act on eachof the two layers, extremely thin, which bound the solid. Thisis always what must be understood by the expression, to maintainthe temperature of the surface constant.

71. We proceed further to examine the case in which thesame solid would be exposed, at one of its faces, to atmosphericair maintained at a constant temperature.

Suppose then that the lower plane preserves the fixed tem-perature a, by virtue of any external cause whatever, and that

54 THEORY OF HEAT, [CHAP. I.

the upper plane, instead of being maintained as formerly at aless temperature b, is exposed to atmospheric air maintainedat that temperature 6, the perpendicular distance of the twoplanes being denoted always by e: the problem is to determinethe final temperatures.

Assuming that in the initial state of the solid, the commontemperature of its molecules is b or less than 6, we can readilyimagine that the heat which proceeds incessantly from the sourceA penetrates the mass, and raises more and more the tempera-tures of the intermediate sections; the upper surface is graduallyheated, and permits part of the heat which has penetrated thesolid to escape into the air. The system of temperatures con-tinually approaches a final state which would exist of itself ifit were once formed; in this final state, which is that whichwe are considering, the temperature of the plane B has a fixedbut unknown value, which we will denote by /3, and since thelower plane A preserves also a permanent temperature a, thesystem of temperatures is represented by the general equation

v = a + z3 v denoting always the fixed temperature of the

section whose height is z. The quantity of heat which flowsduring unit of time across a unit of surface taken on any section

whatever is k , k denoting the interior conducibility.

We must now consider that the upper surface By whosetemperature is /3, permits the escape into the air of a certainquantity of heat which must be exactly equal to that whichcrosses any section whatever L of the solid. If it were not so,the part of the mass included between this section L and theplane B would not receive a quantity of heat equal to thatwhich it loses; hence it would not maintain its state, which iscontrary to hypothesis; the constant flow at the surface is there-fore equal to that which traverses the solid : now, the quantityof heat which escapes, during unit of time, from unit of surfacetaken on the plane B, is expressed by h(/3-b), b being thefixed temperature of the air, and h the measure of the conduci-bility of the surface B; we must therefore have the equation

k = A (/3 - 6), which will determine the value of ft.


From this may be derived a — /3= —^—=-^, an equationtie + k ^

whose second member is known ; for the temperatures a and bare given, as are also the quantities h, h^e.

Introducing this value of a — ft into the general equation

v = a H z, we shall have, to express the temperatures of any

section of the solid, the equation a — v = - ^ ~- , in whichIvU ~T" fir

known quantities only enter with the corresponding variables vand z.

72. So far we have determined the final and permanent stateof the temperatures in a solid enclosed between two infinite andparallel plane surfaces, maintained at unequal temperatures.This first case is, properly speaking, the case of the linear anduniform propagation of heat, for there is no transfer of heat inthe plane parallel to the sides of the solid; that which traversesthe solid flows uniformly, since the value of the flow is the samefor all instants and for all sections.

We will now restate the three chief propositions which resultfrom the examination of this problem; they are susceptible of agreat number of applications, and form the first elements of ourtheory.

1st. If at the two extremities of the thickness e of the solidwe erect perpendiculars to represent the temperatures a and bof the two sides, and if we draw the straight line which joinsthe extremities of these two first ordinates, all the intermediatetemperatures will be proportional to the ordinates of this straight

line; they are expressed by the general equation a-v = z,

v denoting the temperature of the section whose height is z.

2nd. The quantity of heat which flows uniformly, duringunit of time, across unit of surface taken on any section whateverparallel to the sides, all other things being equal, is directlyproportional to the difference a - b of the extreme temperatures,and inversely proportional to the distance e which separates

these sides. The quantity of heat is expressed by K -j-, or

5G THEORY OF HEAT. [CHAP. I.

- K —, if we derive from the general equation the value ofdz

T which is constant; this uniform flow may always be repre-dzsented, for a given substance and in the solid under examination,by the tangent of the angle included between the perpendiculare and the straight line whose ordinates represent the tempera-tures.

3rd. One of the extreme surfaces of the solid being submittedalways to the temperature a, if the other plane is exposed to airmaintained at a fixed temperature b; the plane in contact withthe air acquires, as in the preceding case, a fixed temperature f}9

greater than b, and it permits a quantity of heat to escape intothe air across unit of surface, during unit of time, which is ex-pressed by A(/3 — 5), h denoting the external conducibility ofthe plane.

The same flow of heat h(/3 — b) is equal to that which

traverses the prism and whose value is K (a — /3); we have there-

fore the equation h(J3 — b) = K , which gives the value

of /3.

SECTION V.

Law of the permanent temperatures in a prism of smallthickness.

73. We shall easily apply the principles which have justbeen explained to the following problem, very simple in itself,but one whose solution it is important to base on exact theory.

A metal bar, whose form is that of a rectangular parallelo-piped infinite in length, is exposed to the action of a source ofheat which produces a constant temperature at all points of itsextremity A. It is required to determine the fixed temperaturesat the different sections of the bar.

The section perpendicular to the axis is supposed to be asquare whose side % is so small that we may without sensibleerror consider the temperatures to be equal at different pointsof the same section. The air in which the bar is placed is main-

SECT. V.] STEADY .TEMPERATURE IN A BAR. 57

tained at a constant temperature 0, and carried away by acurrent with uniform velocity.

Within the interior of the solid, heat will pass successivelyall the parts situate to the right of the source, and not exposeddirectly to its action; they will be heated more and more, butthe temperature of each point will not increase beyond a certainlimit. This maximum temperature is not the same for everysection; it in general decreases as the distance of the sectionfrom the origin increases : we shall denote by v the fixed tem-perature of a section perpendicular to the axis, and situate at adistance x from the origin A.

Before every point of the solid has attained its highest degreeof heat, the system of temperatures varies continually, and ap-proaches more and more to a fixed state, which is that whichwe consider. This final state is kept up of itself when it hasonce been formed. In order that the system of temperaturesmay be permanent, it is necessary that the quantity of heatwhich, during unit of time, crosses a section made at a distance xfrom the origin, should balance exactly all the heat which, duringthe same time, escapes through that part of the external surfaceof the prism which is situated to the right of the same section.The lamina whose thickness is dx, and whose external surfaceis 8ldx, allows the escape into the air, during unit of time, ofa quantity of beat expressed by Shlv. dx, h being the measure ofthe external conducibility of the prism. Hence taking the in-tegral jShlv . dx from x = 0 to x = oo , we shall find the quantityof heat which escapes from the whole surface of the bar duringunit of time ; and if we take the same integral from x = 0 tox = x, we shall have the quantity of heat lost through the partof the surface included between the source of heat and the sectionmade at the distance x. Denoting the first integral by (7, whosevalue is constant, and the variable value of the second byjShlv.dx; the difference O-fShlv.dx will express the wholequantity of heat which escapes into the air across the part ofthe surface situate to the right of the section. On the otherhand, the lamina of the solid, enclosed between two sectionsinfinitely near at distances x and x + dx, must resemble an in-finite solid, bounded by two parallel planes, subject to fixedtemperatures v and v + dvy since, by hypothesis, the temperature


does not vary throughout the whole extent of the same section.The thickness of the solid is dx, and the area of the section isU2: hence the quantity of heat which flows uniformly, duringunit of time, across a section of this solid, is, according to the

preceding principles, -4 f&^- , k being the specific internal con-&X

ducibility: we must therefore have the equation

dx

whence hi -y-2

dx

74. We should obtain the same result by considering theequilibrium of heat in a single lamina infinitely thin, enclosedbetween two sections at distances x and x + dx. In fact, thequantity of heat which, during unit of time, crosses the first

section situate at distance x, is — 4d2k -j- . To find that whichdx

flows during the same time across the successive section situateat distance x + dx, we must in the preceding expression change xinto x + dx. which gives — il2k.\-y- + d(-T-)\. If we subtract& \_dx \dxjjthe second expression from the first we shall find how muchheat is acquired by the lamina bounded by these two sectionsduring unit of time; and since the state of the lamina is per-manent, it follows that all the heat acquired is dispersed intothe air across the external surface Sldx of the same lamina: nowthe last quantity of heat is Shlvdx: we shall obtain therefore thesame equation

O7 7 7 i 7 2 7 7 fdv\ T CP# 2Jlofdvdx = 4*1 fed - , - , whence -=—9 = -—- v.\dxj dx kl

75. In whatever manner this equation is formed, it isnecessary to remark that the quantity of heat which passes intothe lamina whose thickness is dx, has a finite value, and that

its exact expression is —^fk-j-. The lamina being enclosed

between two surfaces the first of which has a temperature v,

SECT. V.] STEADY TEMPERATURE IN A BAR. 59

and the second a lower temperature v', we see that the quantityof heat which it receives through the first surface depends onthe difference v - v, and is proportional to i t : but this remarkis not sufficient to complete the calculation. The quantity inquestion is not a differential: it has a finite value, since it isequivalent to all the heat which escapes through that' part ofthe external surface of the prism which is situate to the rightof the section. To form an exact idea of it, we must comparethe lamina whose thickness is dx, with a solid terminated bytwo parallel planes whose distance is e, and which are maintainedat unequal temperatures a and b. The quantity of heat whichpasses into such a prism across the hottest surface, is in factproportional to the difference a - J of the extreme temperatures,but it does not depend only on this difference : all other thingsbeing equal, it is less when the prism is thicker, and in general

it is proportional to . This is why the quantity of heat

which passes through the first surface into the lamina, whose

thickness is dx, is proportional to —,— .

We lay stress on this remark because the neglect of it hasbeen the first obstacle to the establishment of tlie theory. Ifwe did not make a complete analysis of the elements of theproblem, we should obtain an equation not homogeneous, and,a fortiori, we should not be able to form tlie equations whichexpress the movement of heat in more complex cases.

It was necessary also to introduce into the calculation thedimensions of the prism, in order that we might not regard, asgeneral, consequences which observation had furnished in a par-ticular case. Thus, it was discovered by experiment that a barof iron, heated at one extremity, could not acquire, at a distanceof six feet from the source, a temperature of one degree (octo-gesimal1); for to produce this effect, it would be necessary forthe heat of the source to surpass considerably the point of fusionof iron; but this result depends on the thickness of the prismemployed. If it had been greater, the heat would have beenpropagated to a greater distance, that is to say, the point ofthe bar which acquires a fixed temperature of one degree is

1 Reaumur's Scale of Temperature. [A. F.]


much more remote from the source when the bar is thicker, allother conditions remaining the same. We can always raise byone degree the temperature of one end of a bar of iron, by heatingthe solid at the other end; we need only give the radius of thebase a sufficient length: which is, we may say, evident, andof which besides a proof will be found in the solution of theproblem (Art. 78).

76. The integral of the preceding equation isl-2h /fit

v = Ae~xsJ w + J ? e w « ,A and B being two arbitrary constants ; now, if we suppose thedistance x infinite, t he value of the tempera tu re v must be

infinitely smal l ; hence the te rm Be"XNn does not exist in the in-Jlk'

t eg ra l : thus the equation v = Ae~xsJ u represents the permanentstate of the solid ; the tempera ture a t the origin is denoted bythe constant A , since tha t is the value of v when x is zero.

This law according to which the tempera tures decreaseis the same as tha t given by e x p e r i m e n t ; several physicistshave observed the fixed tempera tures at different points of ametal bar exposed at its extremity to t he constant action of asource of heat, and they have ascertained t ha t the distancesfrom the origin represent logarithms, and the temperatures thecorresponding numbers.

77.' The numerical value of the constant quot ient of two con-secutive temperatures being determined by observation, we easily

deduce the value of the ratio p for, denoting by vv v^ the tem-

peratures corresponding to the distances x19 cc2, we have

v2 V k oo2-x1 y '

As for the separate values of h and k, they cannot be deter-mined by experiments of this kind: we must observe also thevarying motion of heat.

78. Suppose two bars of the same material and differentdimensions to be submitted at their extremities to the same tem-

SECT. V.] STEADY TEMPERATURE IN A BAR. 61

perature A; let lx be the side of a section in the first bar, and l2

in the second, we shall have, to express the temperatures of thesetwo solids, the equations

Vl = Ae~Xis/^ and v2 = Ae~x^^,

vlf in the first solid, denoting the temperature of a section madeat distance xlf and v2, in the second solid, the temperature of asection made at distance x2.

When these two bars have arrived at a fixed state, the tem-perature of a section of the first, at a certain distance from thesource, will not be equal to. the temperature of a section of thesecond at the same distance from the focus; in order that thefixed temperatures may be equal, the distances must be different.If we wish to compare with each other the distances xt and x2

from the origin up to the points which in the two bars attainthe same temperature, we must equate the second members of

a)2 Ithese equations, and from them we conclude that -\ = j . Thus

the distances in question are to each other as the square roots ofthe thicknesses.

79. If two metal bars of equal dimensions, but formed ofdifferent substances, are covered with the same coating, whichgives them the same external condueibility1, and if they aresubmitted at their extremities to the same temperature, heat willbe propagated most easily and to the greatest distance from theorigin in that which has the greatest condueibility. To comparewith each other the distances xx and x2 from the common originup to the points which acquire the same fixed temperature, wemust, after denoting the respective conducibilities of the twosubstances by kx and k2, write the equation

6 - * W ^ = e - * 2 ^ , whence ^-2 = p .X2 K2

Thus the ratio of the two conducibilities is that of the squaresof the distances from the common origin to the points whichattain the same fixed temperature.

1 Ingenhousz (1789), Sur Us metaux comme conducteurs de la chaleur. Journalde Physique, xsxiv., 68, 380. Gren's Journal der Physik, Bd. i. [A. F.J

80. It is easy to ascertain how much heat flows during unitof time through a section of the bar arrived at its fixed state:

this quantity is expressed by — 4K2-i-, or 4tAj2khF.e ** **, and

if we take its value at the origin, we shall have 4<Aj2khl3 as themeasure of the quantity of heat which passes from the sourceinto the solid during unit of time; thus the expenditure of thesource of heat is, all 'other things being equal, proportional to thesquare root of the cube of the thickness.

We should obtain the same result on taking the integralJ8hlv. dx from x nothing to x infinite.

SECTION VL

On the heating of closed spaces,

81. We shall again make use of the theorems of Article 72in the following problem, whose solution offers useful applications;it consists in determining the extent of the heating of closedspaces.

Imagine a closed space, of any form whatever, to be filled withatmospheric air and closed on all sides, and that all parts of theboundary are homogeneous and have a common thickness ey sosmall that the ratio of the external surface to the internal surfacediffers little from unity. The space which this boundary termi-nates is heated by a source whose action is constant; for example,by means of a surface whose area is cr maintained at a constanttemperature a.

We consider here only the mean temperature of the air con-tained in the space, without regard to the unequal distribution ofheat in this mass of air; thus we suppose that the existing causesincessantly mingle all the portions of air, and make their tem-peratures uniform.

We see first that the heat which continually leaves the sourcespreads itself in the surrounding air and penetrates the mass ofwhich the boundary is formed, is partly dispersed at the surface,

SECT. VI.] HEATING OF CLOSED SPACES. 63

and passes into the external air, which we suppose to be main-tained at a lower and permanent temperature n. The inner air isheated more and more: the same is the case with the solidboundary: the system of temperatures steadily approaches a finalstate which is the object of the problem, and has the property ofexisting by itself and of being kept up unchanged, provided thesurface of the source cr be maintained at the temperature a, andthe external air at the temperature n.

In the permanent state which we wish to determine the airpreserves a fixed temperature m\ the temperature of the innersurface s of the solid boundary has also a fixed value a; lastly, theouter surface s, which terminates the enclosure, preserves a fixedtemperature b less than a, but greater than n. The quantitieso-, a, s, e and n are known, and the quantities m, a and b areunknown.

The degree of heating consists in the excess of the temperaturem over n} the temperature of the external air; this excess evi-dently depends on the area cr of the heating surface and on itstemperature a; it depends also on the thickness e of the en-closure, on the area s of the surface which bounds it, on thefacility with which heat penetrates the inner surface or thatwhich is opposite to i t ; finally, on the specific conducibility ofthe solid mass which forms the enclosure : for if any one of theseelements were to be changed, the others remaining the same, thedegree of the heating would vary also. The problem is to deter-mine how all these quantities enter into the value of m — n.

82. The solid boundary is terminated by two equal surfaces,each of which is maintained at a fixed temperature; everyprismatic element of the solid enclosed between two opposite por-tions of these surfaces, and the normals raised round the contourof the bases, is therefore in the same state as if it belonged to aninfinite solid enclosed between two parallel planes, maintained atunequal temperatures. All the prismatic elements which com-pose the boundary touch along their whole length. The pointsof the mass which are equidistant from the inner surface haveequal temperatures, to whatever prism they belong ; consequentlythere cannot be any transfer of heat in the direction perpendicularto the length of these prisms. The case is, therefore, the same

-as that of which we have already treated, and we must applyto it the linear equations which have been stated in formerarticles.

83. Thus in the permanent state which we are considering,the flow of heat which leaves the surface cr during a unit of time,is equal to that which, during the same time, passes from thesurrounding air into the inner surface of the enclosure; it isequal also to that which, in a unit of time, crosses an inter-mediate section made within the solid enclosure by a surfaceequal and parallel to those which bound this enclosure; lastly,the same flow is again equal to that which passes from the solidenclosure across its external surface, and is dispersed into the air.If these four quantities of flow of heat were not equal, somevariation would necessarily occur in the state of the temperatures,which is contrary to the hypothesis.

The first quantity is expressed by cr (a — m) g, denoting byg the external conducibility of the surface cr, which belongs tothe source of heat.

The second is s (m — a) h, the coefficient h being the measureof the external conducibility of the surface s, which is exposedto the action of the source of heat.

The third is s K, the coefficient K being the measure of

the conducibility proper to the homogeneous substance whichforms the boundary.

The fourth is s(b — n)H, denoting by H the external con-ducibility of the surface s, which the heat quits to be dispersedinto the air. The coefficients h and H may have very unequalvalues on account of the difference of the state of the two surfaceswhich bound the enclosure; they are supposed to be known, asalso the coefficient K: we shall have then, to determine the threeunknown quantities m, a and 6, the three equations:

a((x — m)g = s (in — a) h,

<T{a-m)g = s Ky

a (a - m)g — s (6 — n) K


84. The value of m is the special object of the problem. Itmay be found by writing the equations in the form

s li

adding, we have m - n = (a - m) P,

denoting by P the known quantity - (& + ^ + -^ I;

whence we conclude

85. The result shews how m — ^, the extent of the heating,depends on given quantities which constitute the hypothesis.We will indicate the chief results to be derived from it \

1st. The extent of the heating m - n is directly proportionalto the excess of the temperature of the source over that of theexternal air.

2nd. The value of m — n does not depend on the form of

the enclosure nor on its volume, but only on the ratio - of the

surface from which the heat proceeds to the surface which receivesit; and also on e the thickness of the boundary.

If we double a the surface of the source of heat, the extentof the heating does not become double, but increases accordingto a certain law which the equation expresses.

1 These results were stated by the author in a rather different manner in theextract from his original memoir published in the Bulletin par la Society Philo-matique de Paris, 1818, pp. 1—11. [A. F.]

F. H. 5


3rd. All the specific coefficients which regulate the actionof the heat, that is to say, g, K, H and h, compose, with the

dimension e, in the value of m — n a single element r + jr+ jr>

whose value may be determined by observation.If we doubled e the thickness of the boundary, we should

have the same result as if, in forming it, we employed a sub-stance whose conducibility proper was twice^as great. Thus theemployment of substances which are bad conductors of heatpermits us to make the thickness of the boundary small; the

effect which is obtained depends only on the ratio •«..

4th. If the conducibility K is nothing, we find ra —w = a;that is to say, the inner air assumes the temperature of thesource : the same is the case if H is zero, or h zero. These con-sequences are otherwise evident, since the heat cannot then bedispersed into the external air.

5th. The values of the quantities g, H, h, K and a, whichwe supposed known, may be measured by direct experiments,as we shall shew in the sequel; but in the actual problem, itwill be sufficient to notice the value of m — n which correspondsto given values of a and of a, and this value may be used to

determine the whole coefficient T + %^ + %., by means of the equa-

tion m —w = (a —n)-_p-r(ln—p) in which p denotes the co-s \ s )

efficient sought. We must substitute in this equation, instead

of - and a—n, the values of those quantities, which we supposes

given, and that of m — n which observation will have madeknown. From it may be derived the value of p, and we maythen apply the formula to any number of other cases.

6th. The coefficient H enters into the value of m-n inthe same manner as the coefficient h; consequently the state ofthe surface, or that of the envelope which covers it, producesthe same effect, whether it has reference to the inner or outersurface.

We should have considered it useless to take notice of these

different consequences, if we were not treating here of entirelynew problems, whose results may be of direct use.

86» We know that animated bodies retain a temperaturesensibly fixed, which we may regard as independent of the tem-perature of the medium in which they live. These bodies are,after some fashion, constant sources of heat, just as inflamedsubstances are in which the combustion has become uniform.We may then, by aid of the preceding remarks, foresee andregulate exactly the rise of temperature in places where a greatnumber of men are collected together. If we there observe theheight of the thermometer under given circumstances, we shalldetermine in advance what that height would be, if the numberof men assembled in the same space became very much greater.

In reality, there are several accessory circumstances whichmodify the results, such as the unequal thickness of the partsof the enclosure, the difference of their aspect, the effects whichthe outlets produce, the unequal distribution of heat in the air.We cannot therefore rigorously apply the rules given by analysis;

.nevertheless these rules are valuable in themselves, because theycontain the true principles of the matter: they prevent vaguereasonings and useless or confused attempts.

87. If the same space were heated by two or more sourcesof different kinds, or if the first inclosure were itself containedin a second enclosure separated from the first by a mass of air,we might easily determine in like manner the degree of heatingand the temperature of the surfaces.

If we suppose that, besides the first source a, there is a secondheated surface ir, whose constant temperature is ft, and externalconducibility j , we shall find, all the other denominations beingretained, the following equation:

1 1\

m — n—+

\ ^TTJ / e 1 1

\K + H + hIf we suppose only one source <x, and'if the first enclosure is

itself contained in a second, .<?', Ji, K\ H\ e, representing the5—2


elements of the second enclosure which correspond to those ofthe first which were denoted by s, h, K, H, e\ we shall find,p denoting the temperature of the air which surrounds the ex-ternal surface of the second enclosure, the following equation :

(a-p)P

The quantity P represents

* (9 + 91 +

We should obtain a similar result if we had three or a greaternumber of successive enclosures; and from this we conclude thatthese solid envelopes, separated by air, assist very much in in-creasing the degree of heating, however small their thicknessmay be.

88. To make this remark more evident, we will compare thequantity of heat which escapes from the heated surface, withthat which the same body would lose, if the surface which en-velopes it were separated from it by an interval filled with air.

If the body A be heated by a constant cause, so that itssurface preserves a fixed temperature b, the air being maintainedat a less temperature a, the quantity of heat which escapes intothe air in the unit of time across a unit of surface will beexpressed by h (b — a), h being the measure of the external con-ducibility. Hence in order that the mass may preserve a fixedtemperature by it is necessary that the source, whatever it maybe, should furnish a quantity of heat equal to hS(b—sa), S de-noting the area of the surface of the solid.

Suppose an extremely thin shell to be detached from thebody A and separated from the solid by an interval filled withair; and suppose the surface of the same solid A to be stillmaintained at the temperature b. We see that the air containedbetween the shell and the body will be heated and will takea temperature a greater than a. The shell itself will attaina permanent state and will transmit to the external air whosefixed temperature is a all the heat which the body loses. Itfollows that the quantity of heat escaping from the solid will


be h8(b — a), instead of being hS(b — a)y for we suppose thatthe new surface of the solid and the surfaces which bound theshell have likewise the same external conducibility h. It isevident that the expenditure of the source of heat will be lessthan it was at first. The problem is to determine the exact ratioof these quantities.

89. Let e be the thickness of the shell, m the fixed tempera-ture of its inner surface, n that of its outer surface, and K itsinternal conducibility. We shall have, as the expression of thequantity of heat which leaves the solid through its surface,h8(b-a').

As that of the quantity which penetrates the inner surfaceof the shell, hS (a — m).

As that of the quantity which crosses any section whatever

of the same shell, KS .e

Lastly, as the expression of the quantity which passes throughthe outer surface into the air, hS (n — a).

All these quantities must be equal, we have therefore thefollowing equations:

h(n — a) = h (a — m),

h (n — a) = k(b — a).

If moreover we write down the identical equation

and arrange them all under the forms

n — a = n — a,

he,. Nm — n = -j^ (n — a),

a — 77% = n — a,

b — a = n — a,

we find, on addition,

I - a = (n - a) 3 + - ^ ) .


The quantity of heat lost by the solid was hSifi-a), whenits surface communicated freely with the air, it is now hS (b - a)

or hS (n - a), which is equivalent to hS

The first quantity is greater than the second in the ratio of

In order therefore to maintain at temperature b a solid whosesurface communicates directly to the air, more than three timesas much heat is necessary than would be required to maintainit at temperature b, when its extreme surface is not adherentbut separated from the solid by any small interval whatever filledwith air.

If we suppose the thickness e to be infinitely small, theratio of the quantities of heat lost will be 3, which would alsobe the value if K were infinitely great.

We can easily account for this result, for the heat beingunable to escape into the external air, without penetrating severalsurfaces, the quantity which flows out must diminish as thenumber of interposed surfaces increases; but we should havebeen unable to arrive at any exact judgment in this case, if theproblem had not been submitted to analysis.

90. We have not considered, in the preceding article, theeffect of radiation across the" layer of air which separates thetwo surfaces; nevertheless this circumstance modifies the prob-lem, since there is a portion of heat which passes directly acrossthe intervening air. We shall suppose then, to make the objectof the analysis more distinct, that the interval between the sur-faces is free from air, and that the heated body is covered byany number whatever of parallel laminao separated from eachother.

If the heat which escapes from the solid through its planesuperficies maintained at a temperature b expanded itself freelyin vacuo and was received by a parallel surface maintained ata less temperature a, the quantity which would be dispersed inunit of time across unit of surface would be proportional to (b — a),the difference of the two constant temperatures: this quantity

SECT. VI.] HEATING OF CLOSED SPACES. 7 1

would be represented by H (b — a), H being the value of the rela-tive conducibility which is not the same as h.

The source which maintains the solid in its original state musttherefore furnish, in every unit of time, a quantity of heat equalto H8(b-a).

We must now determine the new value of this expenditurein the case where the surface of the body is covered by severalsuccessive laminae separated by intervals free from air, supposingalways that the solid is subject to the action of any externalcause whatever which maintains its surface at the temperature b.

Imagine the whole system of temperatures to have becomefixed; let m be the temperature of the under surface of the firstlamina which is consequently opposite to that of the solid, let nbe the temperature of the upper surface of the same lamina,e its thickness, and K its specific conducibility; denote also bym19 nlf m2, n2, m3, n3, m4, n4) &c. the temperatures of the underand upper surfaces of the different laminae, and by K, e, the con-ducibility and thickness of the same laminae; lastly, suppose allthese surfaces to be in a state similar to the surface of the solid,so that the value of the coefficient H is common to them.

The quantity of heat which penetrates the under surface ofa lamina corresponding to any suffix i is HS(ni_1—mi)y that which

crosses this lamina is — (mt — ?iz), and the quantity which escapese

from its upper surface is 38 (nt — mi+l). These three quantities,and all those which refer to the other laminae are equal; we maytherefore form the equation by comparing all these quantitiesin question with the first of them, which is HS (b — m j ; we shallthus have, denoting the number of laminae

b -m1 = h — mv

He ,, x

He n- n2 = -g (6 -


Adding these equations, we find

The expenditure of the source of heat necessary to maintainthe surface of the body A at the temperature b is US (b — a),when this surface sends its rays to a fixed surface maintained atthe temperature a. The expenditure is HS (b — mx) when we placebetween the surface of the body A, and the fixed surface maintainedat temperature a, a numberjof isolated laminae; thus the quantityof heat which the source must furnish is very much less in thesecond hypotheses than in the first, and the ratio of the two

quantities is ^ . If we suppose the thickness e of the

laminae to be infinitely small, the ratio is - . The expenditurej

of the source is then inversely as the number of laminae whichcover the surface of the solid.

91. The. examination of these results and of those which weobtained when the intervals between successive enclosures wereoccupied by atmospheric air explain clearly why the separationof surfaces and the intervention of air assist very much in re-taining heat.

Analysis furnishes in addition analogous consequences whenwe suppose the source to be external, and that the heat whichemanates from it crosses successively different diathermanousenvelopes and the air which they enclose. This is what hashappened when experimenters have exposed to the rays of thesun thermometers covered by several sheets of glass within whichdifferent layers of air have been enclosed.

For similar reasons the temperature of the higher regionsof the atmosphere is very much less than at the surface of theearth.

SECT. VII.] MOVEMENT IN THREE DIMENSIONS. 73

In general the theorems concerning the heating of air inclosed spaces extend to a great variety of problems. It wouldbe useful to revert to them when we wish to foresee and regulatetemperature with precision, as in the case of green-houses, drying-houses, sheep-folds, work-shops, or in many civil establishments,such as hospitals, barracks, places of assembly.

In these different applications we must attend to accessorycircumstances which modify the results of analysis, such as theunequal thickness of different parts of the enclosure, the intro-duction of airr &c.; but these details would draw us away fromour chief object, which is the exact demonstration of generalprinciples.

For the rest,, we have considered only, in what has just beensaid, the permanent state of temperature in closed spaces. Wecan in addition express analytically the variable state whichprecedes, or that which begins to take place when the source ofheat is withdrawn, and we can also ascertain in this way, howthe specific properties of the bodies which we employ, or theirdimensions affect the progress and duration of the heating ; butthese researches require a different analysis, the principles ofwhich will be explained in the following chapters.

SECTION VIL

On the uniform movement of heat in three dimensions.

92. Up to this time we have considered the uniform move-ment of heat in one dimension only, but it is easy to apply thesame principles to the case in which heat is propagated uniformlyin three directions at right angles.

Suppose the different points of a solid enclosed by six planesat right angles to have unequal actual temperatures representedby the linear equation v = A + ax + by + cz, x, y> z, being therectangular co-ordinates of a molecule whose temperature is v.Suppose further that any external causes whatever acting on thesix faces of the prism maintain every one of the molecules situatedon the surface, at its actual temperature expressed by the general

equationv = A + ax + by + cz (a),


we shall prove that the same causes which, by hypothesis, keepthe outer layers of the solid in their initial state, are sufficientto preserve also the actual temperatures of every one of the innermolecules, so that their temperatures do not cease to be repre-sented by the linear equation.

The examination of this question is an element of thegeneral theory, it will serve to determine the laws of the variedmovement of heat in the interior of a solid of any form whatever,for every one of the prismatic molecules of which the body iscomposed is during an infinitely small time in a state similarto that which the linear equation (a) expresses. We may then,by following the ordinary principles of the differential calculus,easily deduce from the notion of uniform movement the generalequations of varied movement.

93. In order to prove that when the extreme layers of thesolid preserve their temperatures no change can happen in theinterior of the mass, it is sufficient to compare with each otherthe quantities of heat which, during the same instant, cross twoparallel planes.

Let b be the perpendicular distance of these two.planes whichwe first suppose parallel to the horizontal plane of x and y. Letm and m be two infinitely near molecules, one of which is abovethe first horizontal plane and the other below i t : let x, yy z bethe co-ordinates of the first molecule, and x, y, z those of thesecond. In like manner let M and Mr denote two infinitelynear molecules, separated by the second horizontal plane andsituated, relatively to that plane, in the same manner as m andm are relatively to the first plane; that is to say, the co-ordinatesof M are x, y, z + 6, and those of M are x, y\ z + b. It is evidentthat the distance mm of the two molecules m and m' is equalto the distance MM of the two molecules M and M ; further,let v be the temperature of m, and v that of m, also let V andV be the temperatures of M and M\ it is easy to see that thetwo differences v - v and V- V are equal; in fact, substitutingfirst the co-ordinates of m and m in the general equation

v = A + ax + by + cz,

we find v - v = a (x - x) 4- b (y - y) + c (z - / ) ,

SECT. VII.] MOVEMENT IN THREE DIMENSIONS. 75

and then substituting the co-ordinates of M and M\ we find alsoV- V = a (x - x) +b{y-y)+c(z- z). Now the quantity ofheat which m sends to m depends on the distance mm, whichseparates these molecules, and it is proportional to the differencev — v of their temperatures. This quantity of heat transferredmay be represented by

q (v — vf) dt;

the value of the coefficient q depends in some manner on thedistance mm, and on the nature of the substance of which thesolid is formed, dt is the duration of the instant. The quantityof heat transferred from M to M', or the action of M on M' isexpressed likewise by q (F — V) dt, and the coefficient q is thesame as in the expression q (v - v) dt, since the distance MM' isequal to mm and the two actions are effected in the same solid:furthermore V— V is equal to v — v, hence the two actions areequal.

If we choose two other points n and n, very near to eachother, which transfer heat across the first horizontal plane, weshall find in the same manner that their action is equal to thatof two homologous points N and N' which communicate heatacross the second horizontal plane. We conclude then that thewhole quantity of heat which crosses the first plane is equal tothat which crosses the second plane during the same instant.We should derive the same result from the comparison of twoplanes parallel to the plane of x and z, or from the comparisonof two other planes parallel to the plane of y and z. Henceany part whatever of the solid enclosed between six planes atright angles, receives through each of its faces as much heat asit loses through the opposite face; hence no portion of the solidcan change temperature.-

94. From this we see that, across one of the planes inquestion, a quantity of heat flows which is the same at all in-stants, and which is also the same for all other parallel sections.

In order to determine the value of this constant flow weshall compare it with the quantity of heat which flows uniformlyin the most simple case, which has been already discussed. Thecase is that of an infinite solid enclosed between two infinite


planes and maintained in a constant state. We have seen thatthe temperatures of the different points of the mass are in thiscase represented by the equation v= A + cz; we proceed to provethat the uniform flow of heat propagated in the vertical directionin the infinite solid is equal to that which flows in the samedirection across the prism enclosed by six planes at right angles.This equality necessarily exists if the coefficient c in the equationv = A + cz, belonging to the first solid, is the same as the coeffi-cient c in the more general equation v = A + ax + by + cz whichrepresents the state of the prism. ID fact, denoting by H aplane in this prism perpendicular to z, and by m and fi twomolecules very near to each other,, the first of which m is belowthe plane Hy and the second above this plane, let v be thetemperature of m whose co-ordinates are x, y, z, and w thetemperature of /u whose co-ordinates are x +• a, y + /3, z-\- 7. Takea third molecule JUL whose co-ordinates are x — a, y — /3, z + y, andwhose temperature may be denoted by w\ We see that /JL andJJ! are on the same horizontal plane, and that the vertical drawnfrom the middle point of the line /JL/JL, which joins these twopoints, passes through the point m, so that the distances m/x andTiifju are equal. The action of m on fi, or the quantity of heatwhich the first of these molecules sends to the other across theplane H; depends on the difference v - w of their temperatures.The action of m on / / depends in the same manner on thedifference v — w of the temperataes of these molecules, sincethe distance of m from fi is the same as that of m from p'. Thus,expressing by q (v - w) the action of m on fju during the unit oftime, we shall have q (y — w) to express the action of m on //,q being a common unknown factor, depending on the distancem/jb and on the nature of the solid. Hence the sum of tha twoactions exerted during unit of time is q (y.— w + v — w).

If instead of x, y> and z, in the general equation

v = A + ax + by + cz,

we substitute the co-ordinates of m and then those of fi and fi\we shall find

v — w = — aoc — 6/3 — cy,

— cy.

SECT. VII. ] MOVEMENT IN THREE DIMENSIONS. 77

The sum of the two actions of on on /JL and of m on jjf is there-fore — 2qcy.

Suppose then that the plane H belongs to the infinite solidwhose temperature equation is v = A -t- cz, and that we denotealso by my /JL and /// those molecules in this solid whose co-ordinates are x, y, z for the first, x + a, y + ft, z + 7 for the second,and x — a, y — ft, z + 7 for the third : we shall have, as in thepreceding case, v- w-\-v — w'= — 2cy. Thus the sum of the twoactions of m on \x and of m on /-&', is the same in the infinite solidas in the prism enclosed between the six planes at right angles.

We should obtain a similar result, if we considered the actionof another point n below the plane H on two others v and v\situated at the same height above the plane. Hence, the sumof all the actions of this kind, which are exerted across the planeH, that is to say the whole quantity of heat which, during unitof time, passes to the upper side of this surface, by virtue of theaction of very near molecules which it separates, is always thesame in both solids.

95. In the second of these two bodies, that which is boundedby two infinite planes, and whose temperature equation isv =A-\- cz, we know that the quantity of heat which flows duringunit of time across unit of area taken on any horizontal sectionwhatever is — cK, c being the coefficient of z, and K the specificconducibility; hence, the quantity of heat which, in the prism,enclosed between six planes at right angles, crosses during unitof time, unit of area taken on any horizontal section whatever,is also — cK, when the linear equation which represents the tem-peratures of the prism is

v = A + ax + hy + cz.

In the same way it may be proved that the quantity of heatwhich, during unit of time, flows uniformly across unit of areataken on any section whatever perpendicular to xy is expressedby - aK} and that the whole quantity which, during unit of time,crosses unit of area taken on a section perpendicular to y, isexpressed by - bK.

The theorems which we have demonstrated in this and thetwo preceding articles, suppose the direct action of heat in the

78 THEORY OF HEAT. [CHAP. .1.

interior of the mass to be limited to an extremely small distance,but they would still be true, if the rays of heat sent out by eachmolecule could penetrate directly to a quite appreciable distance,but it would be necessary in this case, as we have remarked inArticle 70, to suppose that the cause which maintains the tem-peratures of the faces of the solid affects a part extending withinthe mass to a finite depth.

SECTION VIII.

Measure of the movement of heat at a given point of a solid mass.

96. It still remains for us to determine one of the principalelements of the theory of heat, which consists in defining and inmeasuring exactly the quantity of heat which passes throughevery point of a solid mass across a plane whose direction is given.

If heat is unequally distributed amongst the molecules of thesame body, the temperatures at any point will vary every instant.Denoting by t the time which has elapsed, and by v the tem-perature attained after a time t by an infinitely small moleculewhose co-ordinates are x,y,z\ the variable state of the solid will beexpressed by an equation similar to the following v = F(x, y, z, t).Suppose the function F to be given, and that consequently wecan determine at every instant the temperature of any pointwhatever; imagine that through the point m we draw a hori-zontal plane parallel to that of x and y, and that on this planewe trace an infinitely small circle a>, whose centre is at m ; it isrequired to determine what is the quantity of heat which duringthe instant dt will pass across the circle co from the part of thesolid which is below the plane into the part above it.

All points extremely near to the point m and under the planeexert their action during the infinitely small instant dt} on allthose which are above the plane and extremely near to the pointm, that is to say, each of the points situated on one side of thisplane will send heat to each of those which are situated on theother side.

We shall consider as positive an action whose effect is totransport a certain quantity of heat above the plane, and asnegative that which causes heat to pass below the plane. The

SECT. VIII.] MOVEMENT IN A SOLID MASS. 79

sum of all the partial actions which are exerted across the circleco, that is to say the sum of all the quantities of heat which,crossing any point whatever of this circle, pass from the partof the solid below the plane to the part above, compose the flowwhose expression is to be found.

It is easy to imagine that this flow may not be the samethroughout the whole extent of the solid, and that if at anotherpoint m we traced a horizontal circle co equal to the former, thetwo quantities of heat which rise above these planes co and cof

during the same instant might not be equal: these quantities arecomparable with each other and their ratios are numbers whichmay be easily determined.

97. We know already the value of the constant flow for thecase of linear and uniform movement; thus in the solid enclosed be-tween two infinite horizontal planes, one of which is maintained atthe temperature a and the other at the temperature b, the flow ofheat is the same for every part of the mass; we may regard it astaking place in the vertical direction only. The value correspond-ing to unit of surface and to unit of time is K ( ) , e denoting

the perpendicular distance of the two planes, and K the specificconducibility: the temperatures at the different points of the

solid are expressed by the equation v = a — I j z.

When the problem is that of a solid comprised between sixrectangular planes, pairs of which are parallel, and the tem-peratures at the different points are expressed by the equation

v = A + ax + by + cz,the propagation takes place at the same time along the directionsof x, of 2/, of z; the quantity of heat which flows across a definiteportion of a plane parallel to that of x and y is the same through-out the whole extent of the prism; its value corresponding to unitof surface, and to unit of time is - cK, in the direction of z, it is— bK, in the direction of y, and - aK in that of x.

In general the value of the vertical flow in the two cases whichwe have just cited, depends only on the coefficient of z and on

dvthe specific conducibility K\ this value is always equal to ~K^ •


The expression of the quantity of heat which, during the in-stant dt, flows across a horizontal circle infinitely small, whose areais co, and passes in this manner from the part of the solid which isbelow the plane of the circle to the part above, is, for the two cases

in question, — K -j- codt

98. It is easy now to generalise this result and to recognisethat it exists in every case of the varied movement of heat ex-pressed by the equation v = F (x, y, z, t).

Let us in fact denote by x, y, z, the co-ordinates of this pointm, and its actual temperature by-?/. Let x 4- f, y + r], s' -f £, bethe co-ordinates of a point /JL infinitely near to the point m, andwhose temperature is w; £ ??, f are quantities infinitely small addedto the co-ordinates x', y, z ; they determine the position ofmolecules infinitely near to the point m, with respect to threerectangular axes, whose origin is at ra, parallel to the axes ofx,y} and z. Differentiating the equation

v = / (x> V> z> 0and replacing the differentials by f, rj, %, we shall have, to expressthe value of w which is equivalent to v + dv} the linear equation

, dv y dv dv\ , re - ± f dv dv dv rw = v + -j-s + ~T-r) + -j-±> ^ e coefficients v9-j-9 -7-,—r > are func-dx* dy dz*' dx dydz'tions of ^, y, z, t, in which the given and constant values x, y\ z,which belong to the point m, have been substituted for x, y, z.

Suppose that the same point in belongs also to a solid enclosedbetween six rectangular planes, and that the actual temperaturesof the points of this prism, whose dimensions are finite, are ex-pressed by the linear equation w = A +a^ + brj + c£; and thatthe molecules situated on the faces which bound the solid aremaintained by some external cause at the temperature which isassigned to them by the linear equation. £ TJ, % are the rectangularco-ordinates of a molecule of the prism, whose temperature is w,referred to three axes whose origin is at m.

This arranged, if we take as the values of the constant coeffi-cients A, a, by c, which enter into the equation for the prism, the

, dv dv dv . . . _ _

quantities v, 2x'dy'"dz' w h l c h b e l o r i g t o the differential equa-

tion ; the state of the prism expressed by the equation

SECT. VIII.] MOVEMENT IN A SOLID MASS. 81

dx dy dz

will coincide as nearly as possible with the state of the solid; thatis to say, all the molecules infinitely near to the point m will havethe same temperature, whether we consider them to be in the solidor in the prism. This coincidence of the solid and the prism isquite analogous to that of curved surfaces with the planes whichtouch them.

It is evident, from this, that the quantity of heat which flowsin the solid across the circle &>, during the instant dt, is the sameas that which flows in the prism across the same circle; for all themolecules whose actions concur in one effect or the other, havethe same temperature in the two solids. Hence, the flow in

dvquestion, in one solid or the other, is expressed by — K -7- codt.

azIt would be — K ^- codt, if the circle co, whose centre is m. were

dydv

perpendicular to the axis of y, and — K -y- wdt, if this circle were

perpendicular to the axis of x.The value of the flow which we have just determined varies

in the solid from one point to another, and it varies also withthe time. We might imagine it to have, at all the points of aunit of surface, the same value as at the point m, and to preservethis value during unit of time; the flow would then be expressed

by— K-r/it would b e - i T - r in the direction of y, and — K-j~J dz dij u dxin that of x. We shall ordinarily employ in calculation thisvalue of the flow thus referred to unit of time and to unit ofsurface.

99. This theorem serves in general to measure the velocitywith which heat tends to traverse a given point of a planesituated in any manner whatever in the interior of a solid whosetemperatures vary with the time. Through the given point m,a perpendicular must be raised upon the plane, and at everypoint of this perpendicular ordinates must be drawn to representthe actual temperatures at its different points. A plane curvewill thus be formed whose axis of abscissae is the perpendicular.

F. H. 6


The fluxion of the ordinate of this curve, answering to the pointm, taken with the opposite sign, expresses the velocity withwhich heat is transferred across the plane. This fluxion of theordinate is known to be the tangent of the angle formed bythe element of the curve with a parallel to the abscissae.

The result which we have just explained is that of whichthe most frequent applications have been made in the theoryof heat. We cannot discuss the different problems withoutforming a very exact idea of the value of the flow at every pointof a body whose temperatures are variable. It is necessary toinsist on this fundamental notion; an example which we areabout to refer to will indicate more clearly the use which hasbeen made of it in analysis.

100. Suppose the different points of a cubic mass, an edgeof which has the length ir, to have unequal actual temperaturesrepresented by the equation v = cos x cos y cos z. The co-ordinates xy y, z are measured on three rectangular axes, whoseorigin is at the centre of the cube, perpendicular to the faces.The points of the external surface of the solid are at the actualtemperature 0, and it is supposed also that external causesmaintain at all these points the actual temperature 0. On thishypothesis the body will be cooled more and more, the tem-peratures of all the points situated in the interior of the masswill vary, and, after an infinite time, they will all attain thetemperature 0 of the surface. Now, we shall prove in the sequel,that the variable state of this solid is expressed by the equation

v = e~9t cos x cos y cos z>

SKthe coefficient g is equal to 7j—j-y K is the specific conduci-bility of the substance of which the solid is formed, D is thedensity and G the specific heat; t is the time elapsed.

We here suppose that the truth of this equation is admitted,and we proceed to examine the use which may be made of itto find the quantity of heat which crosses a given plane parallelto one of the three planes at the right angles.

If, through the point in, whose co-ordinates are x9 y} z} wedraw a plane perpendicular to z9 we shall find, after the mode

SECT. VIII.] MOVEMENT IN A CUBE. 83

of the preceding article, that the value of the flow, at this pointrln

and across the plane, is — K -j-, or Re'9* cos x. cos y . sin z. The

quantity of heat which, daring the instant dt, crosses an infinitelysmall rectangle, situated on this plane, and whose sides aredx and dy, is

K e~gt cos x cos y sin z dx dy dt.

Thus the whole heat which, during the instant dt, crosses theentire area of the same plane, is

K e~gt sin z . dt j I cos x cos ydxdy;

the double integral being taken from x = — ~ IT up to x = ~ TT,

and from y = — - IT up to y = - ir. We find then for the ex-

pression of this total heat,

4< Ke~gt sin z.dt.

If then we take the integral with respect to t, from t = 0 tot = t, we shall find the quantity of heat which has crossed thesame plane since the cooling began up to the actual moment.

4iTThis integral is — sin z(l— e~9t), its value at the surface is

so that after an infinite time the quantity of heat lost through4>K

one of the faces is — . The same reasoning being applicable»y

to each of the six faces, we conclude that the solid has lost by its24 K

complete cooling a total quantity of heat equal to —— or 8GD,since g is equivalent to -frh • ^he total heat which is dissipated

during the cooling must indeed be independent of the specialconducibility K, which can only influence more or less thevelocity of cooling.

6—2

84 THEORY OF HEAT. [CH. I. SECT. VIII.

100. A. We may determine in another manner the quantityof heat which the solid loses during a given time, and this willserve in some degree to verify the preceding calculation. Infact, the mass of the rectangular molecule whose dimensions aredx, dy, dz, is Ddxdydz, consequently the quantity of heatwhich must be given to it to bring it from the temperature 0 tothat of boiling water is CDdxdydz, and if it were required toraise this molecule to the temperature v, the expenditure of heatwould be v CD dx dy dz.

It follows from this, that in order to find the quantity by

which the heat of the solid, after time t, exceeds that which

it contained at the temperature 0, we must take the mul-

tiple integral jijv CD dx dy dz, between the limits x = — - IT,

1 1 1 1 1

We thus find, on substituting for v its value, that is to say

e~9t cos x cos y cos z>

that the excess of actual heat over that which belongs to thetemperature 0 is 8 CD (1 - e'9t); or, after an infinite time,8 CD, as we found before.

We have described, in this introduction, all the elements whichit is necessary to know in order to solve different problemsrelating to the movement of heat in solid bodies, and we havegiven some applications of these principles, in order to shewthe mode of employing them in analysis ; the most importantuse which we have been able to make of them, is to deducefrom them the general equations of the propagation of heat,which is the subject of the next chapter.

Note on Art. 76. The researches of J. D. Forbes on the temperatures of a longiron bar heated at one end shew conclusively that the conducting power K is not con-stant, but diminishes as the temperature increases.—Transactions of the RoyalSociety of Edinburgh, Vol. xxm. pp. 133—146 and Vol. xxiv. pp. 73—110.

Note on Art. 98. General expressions for the flow of heat within a mass inwhich the conductibility varies with the direction of the flow are investigated byLame in his Theorie Analytique de la Chaleur, pp. 1—8. [A. F.J

CHAPTER II.

EQUATIONS OF THE MOVEMENT OF HEAT.

SECTION I.

Equation of the varied movement of heat in a ring.

101. W E might form the general equations which representthe movement of heat in solid bodies of any form whatever, andapply them to particular cases. But this method would ofteninvolve very complicated calculations which may easily be avoided.There are several problems which it is preferable to treat in aspecial manner by expressing the conditions which are appropriateto them; we proceed to adopt this course and examine separatelythe problems which have been enunciated in the first section ofthe introduction; we will limit ourselves at first to forming thedifferential equations, and shall give the integrals of them in thefollowing chapters.

102. We have already considered the uniform movement ofheat in a prismatic bar of small thickness whose extremity isimmersed in a constant- source of heat. This first case offered nodifficulties, since there was no reference except to the permanentstate of the temperatures, and the equation which expresses themis easily integrated. The following problem requires a more pro-found investigation; its object is to determine the variable stateof a solid ring whose different points have received initial tempe-ratures entirely arbitrary.

The solid ring or armlet is generated by the revolution ofa rectangular section about an axis perpendicular to the plane of

86 THEORY OF HEAT. [CHAP. II.

the ring (see figure 3), I is the perimeter of the section whose area. is S, the coefficient h measures the external con-

ducibility, K the internal conducibility, C thespecific capacity for heat, D the density. The lineoxxx" represents the mean circumference of thearmlet, or that line which passes through thecentres of figure of all the sections; the distanceof a section from the origin o is measured by the

arc whose length is x\ R is the radius of the mean circumference.It is supposed that on account of the small dimensions and of

the form of the section, we may consider the temperature at thedifferent points of the same section to be equal.

103. Imagine that initial arbitrary temperatures have beengiven to the different sections of the armlet, and that the solid isthen exposed to air maintained at the temperature 0, and dis-placed with a constant velocity; the system of temperatures willcontinually vary, heat will be propagated within the ring, anddispersed at the surface: it is required to determine what will bethe state of the solid at any given instant.

Let v be the temperature which the section situated at distancex will have acquired after a lapse of time t; v is a certain functionof x and t, into which all the initial temperatures also must enter:this is the function which is to be discovered.

104. We will consider the movement of heat in an infinitelysmall slice, enclosed between a section made at distance x andanother section made at distance x + dx. The state of this slicefor the duration of one instant is that of an infinite solid termi-nated by two parallel planes maintained at unequal temperatures;thus the quantity of heat which flows during this instant dt acrossthe first section, and passes in this way from the part of the solidwhich precedes the slice into the slice itself, is measured accordingto the principles established in the introduction, by the product offour factors, that is to say, the conducibility K, the area of the

section 8, the ratio — -j-, and the duration of the instant; its

expression is —KS-r-dt. To determine the quantity of heat

SECT. I.] VARIED MOVEMENT IN A RING. 87

which escapes from the same slice across the second section, andpasses into the contiguous part of the solid, it is only necessaryto change x into x + dx in the preceding expression, or, which isthe same thing, to add to this expression its differential takenwith respect to x; thus the slice receives through one of its faces

a quantity of heat equal to — KS-j-dt, and loses through the

opposite face a quantity of heat expressed by

- KS^ dt - KS ^ dx dt.ax ax

It acquires therefore by reason of its position a quantity of heatequal to the difference of the two preceding quantities, that is

KS~V2dxdt.

dx2

On the other hand, the same slice, whose external surface isIdx and whose temperature differs infinitely little from v, allowsa quantity of heat equivalent to hlvdxdt to escape into the airduring the instant dt; it follows from this that this infinitelysmall part of the solid retains in reality a quantity of heat

d2vrepresented by KS -^ dx dt — hlv dx dt which makes its tempe-

rature vary. The amount of this change must be examined.

105. The coefficient G expresses how much heat is requiredto raise unit of weight of the substance in question from tempe-rature 0 up to temperature 1; consequently, multiplying thevolume Sdx of the infinitely small slice by the density Dy toobtain its weight, and by G the specific capacity for heat, we shallhave CD Sdx as the quantity of heat which would raise thevolume of the slice from temperature 0 up to temperature 1.Hence the increase of temperature which results from the addition

d2vof a quantity of heat equal to KS -f-^dxdt- hlv dx dt will be

found by dividing the last quantity by CD Sdx. Denoting there-fore, according to custom, the increase of temperature which takes

place during the instant dt by _, dt, we shall have the equation


dv K d2v hidt CD dx* CDS' .(&)

We shall explain in the sequel the use which may be made ofthis equation to determine the complete solution, and what thedifficulty of the problem consists in; we limit ourselves here toa remark concerning the permanent state of the armlet.

106. Suppose that, the plane of the ring being horizontal,sources of heat, each of which exerts a constant action, are placedbelow different points m, n> p, q etc.; heat will be propagated inthe solid, and that which is dissipated through the surface beingincessantly replaced by that which emanates from the sources, thetemperature of every section of the solid will approach more andmore to a stationary value which varies from one section toanother. In order to express by means of equation (b) the law ofthe latter temperatures, which would exist of themselves if theywere once established, we must suppose that the quantity v does

not vary with respect to t\ wThich annuls the term -3-. We thus

have the equation

—— = v, whence v = Me~x KS + Ne KS,

M and N being two constants1.

1 This equation is the same as the equation for the steady temperature of afinite bar heated at one end (Art. 76), except that I here denotes the perimeter ofa section whose area is S. In the case of the finite bar we can determine tworelations between the constants M and N: for, if V be the temperature at thesource, where x = 0, V=M+N; and if at the end of the bar remote from the source,where x = L suppose, we make a section at a distance dx from that end, the flow

through this section is, in unit of time, -ES^- , and this is equal to the waste

of heat through the periphery and free end of the slice, hv(ldx + 8) namely;hence ultimately, dx vanishing,

hv + K — = 09 when x = L,dx y '

that is

Me~Lf* KS+Ne +L^ KS= \J ^ f Me " ^ *s - Ne +L**

Cf. Verdet, Conferences de Physique, p. 37. [A. F.]

SECT. I.] STEADY MOVEMENT IN A RING. 80

107. Suppose a portion of the circumference of the ring,situated between two successive sources of heat, to be dividedinto equal parts, and denote by vv v2> v3> vv &c, the temperaturesat the points of division whose distances from the origin arexv xv x& xv &c«; the relation between v and x will be giyen bythe preceding equation, after that the two constants have beendetermined by means of the two values of v corresponding to

the sources of heat. Denoting by a the quantity e KS, andby X the distance x2 —xx of two consecutive points of division,we shall have the equations:

vx = Ma** + NoT\

v2 = MOLK . ofi + JVjf V*1,

v3 =

whence we derive the following relation — 8 = aA + a~\

We should find a similar result for the three points whosetemperatures are v2, v3, v4, and in general for any three consecutivepoints. It follows from this that if we observed the temperaturesvv v2> v3, v4, v5 &c. of several successive points, all situated betweenthe same two sources m and w and separated by a constantinterval X, we should perceive that any three consecutive tempe-ratures are always such that the sum of the two extremes dividedby the mean gives a constant quotient ax+ a~\

108. If, in the space included between the next two sources ofheat n and p, the temperatures of other different points separatedby the same interval X were observed, it would still be found thatfor any three consecutive points, the sum of the two extremetemperatures, divided by the mean, gives the same quotienta* _j- a - \ The value of this quotient depends neither on theposition nor on the intensity of the sources of heat.

109. Let q be this constant value, we have the equation

we see by this that when the circumference is divided into equalparts, the temperatures at the points of division, included between

90 THEORY OF HEAT. [CHAP. II;

two consecutive sources of heat, are represented by the terms ofa recurring series whose scale of relation is composed of two termsq and — 1.

Experiments have fully confirmed this result. We have ex-posed a metallic ring to the permanent and simultaneous actionof different sources of heat, and we have observed the stationarytemperatures of several points separated by constant intervals; wealways found that the temperatures of any three consecutivepoints, not separated by a source of heat, were connected by therelation in question. Even if the sources of heat be multiplied,and in whatever manner they be disposed, no change can be

effected in the numerical value of the quotient -* 3; it depends

only on the dimensions or on the nature of the ring, and not onthe manner in which that solid is heated.

110. When we have found, by observation, the value of the

constant quotient q or 1 a

3 , the value of aA may be derived

from it by means of the equation aA + oTx = q. One of the rootsis ax, and other root is a~\ This quantity being determined,

we may derive from it the value of the ratio ^ . , which isXL

y (log a)2. Denoting aA by co, we shall have co2 — qco + 1 = 0. Thus

the ratio of the two conducibilities is found by multiplying —V

by the square of the hyperbolic logarithm of one of the roots ofthe equation co2 — qco + 1 = 0, and dividing the product by \2.

SECTION II.

Equation of the varied movement of heat in a solid sphere,

111. A solid homogeneous mass, of the form of a sphere,having been immersed for an infinite time in. a medium main-tained at a permanent temperature 1, is then exposed to air whichis kept at temperature 0, and displaced with constant velocity:it is required to determine the successive states of the body duringthe whole time of the cooling.

SECT. II.] VARIED MOVEMENT IN A SPHERE. 91

Denote by x the distance of any point whatever from thecentre of the sphere, and by v the temperature of the same point,after a time t has elapsed ; and suppose, to make the problemmore general, that the initial temperature, common to all pointssituated at the distance x from the centre, is different for differentvalues of x; which is what would have been the case if the im-mersion had not lasted for an infinite time.

Points of the solid, equally distant from the centre, will notcease to have a common temperature; v is thus a function of xand t. When we suppose t = 0, it is essential that the value ofthis function should agree with the initial state which is given,and which is entirely arbitrary.

112. We shall consider the instantaneous movement of heatin an infinitely thin shell, bounded by two spherical surfaces whoseradii are x and x + dx: the quantity of heat which, during aninfinitely small instant dt, crosses the lesser surface whose radiusis x, and so passes from that part of the solid which is nearest tothe centre into the spherical shell, is equal to the product of fourfactors which are the conducibility K, the duration dt, the extent

dvkirx* of surface, and the ratio - ^ , taken with the negative sign ;

it is expressed by — kKirx* -j- dt.

To determine the quantity of heat which flows during thesame instant through the second surface of the same shell, andpasses from this shell into the part of the solid which envelops it,x must be changed into x -f dx, in the preceding expression : that

el i)

is to say, to the term - 4 Z W -j- dt must be added the differen-

tial of this term taken with respect to x. We thus find

- ±Kira? d^dt- kKird U ^) . dtax \ ax]

as the expression of the quantity of heat which leaves the spheri-cal shell across its second surface; and if we subtract this quantityfrom that which enters through the first surface, we shall have

(x2 -^-] dt This difference is evidently the quantity ofV dx)

heat which accumulates in the intervening shell, and whose effectis to vary its temperature.

113. The coefficient G denotes the quantity of heat which isnecessary to raise, from temperature 0 to temperature 1, a definiteunit of weight; D is the weight of unit of volume, 4vrx2dx is thevolume of the intervening layer, differing from it only by aquantity which may be omitted: hence ^irCDx^dx is the quantityof heat necessary to raise the intervening shell from temperature0 to temperature 1. Hence it is requisite to divide the quantityof heat which accumulates in this shell by 4<7rCDx2dx, and weshall then find the increase of its temperature v during the timedt. We thus obtain the equation

Kd (x2-^)V dx)

x dx

_ i (dS_ 2 dv\o r dt~CD* \drf + x dx)

114. The preceding equation represents the law of the move-ment of heat in the interior of the solid, but the temperatures ofpoints in the surface are subject also to a special condition whichmust be expressed. This condition relative to the state of thesurface may vary according to the nature of the problems dis-cussed : we may suppose for example, that, after having heatedthe sphere, and raised all its molecules to the temperature ofboiling water, the cooling is effected by giving to all points in thesurface the temperature 0, and by retaining them at this tem-perature by any external cause whatever. In this case we mayimagine the sphere, whose variable state it is desired to determine,to be covered by a very thin envelope on which the cooling agencyexerts its action. It may be supposed, 1°, that this infinitelythin envelope adheres to the solid, that it is of the same substanceas the solid and that it forms a part of it, like the other portionsof the mass; 2°, that all the molecules of the envelope are sub-jected to temperature 0 by a cause always in action which preventsthe temperature from ever being above or below zero. To expressthis condition theoretically, the function v, which contains x and t,

SECT. II.] VAEIED MOVEMENT IN A SPHERE. 93

must be made to become nul, when we give to x its completevalue X equal to the radius of the sphere, whatever else the valueof t may be. We should then have, on this hypothesis, if wedenote by <j> (a?, t) the function of x and t, which expresses thevalue of v, the two equations

dvK^ fd2v . 2 dvy

dt CDFurther, it is necessary that the initial state should be repre-sented by the same function <£ (x, t) : we shall therefore have as asecond condition (f> (#, 0) = 1. Thus the variable state of a solidsphere on the hypothesis which we have first described will berepresented by a function v, which must satisfy the three precedingequations. The first is general, and belongs at every instant toall points of the mass ; the second affects only the molecules atthe surface, and the third belongs only to the initial state.

115/ If the solid is being cooled in air, the second equation isdifferent; it must then be imagined that the very thin envelopeis maintained by some external cause, in a state such as to pro-duce the escape from the sphere, at every instant, of a quantity ofheat equal to that which the presence of the medium can carryaway from it.

Now the quantity of heat which, during an infinitely small"

instant dt, flows within the interior of the solid across the spheri-

cal surface situate at distance x, is equal to — 4<KTTX2 -=- dt; andctoc

this general expression is applicable to all values of x. Thus, bysupposing x = X we shall ascertain the quantity of heat which inthe variable state of the sphere would pass across the very thinenvelope which bounds i t ; on the other hand, the external surfaceof the solid having a variable temperature, which we shall denoteby F, would permit the escape into the air of a quantity of heatproportional to that temperature, and to the extent of the surface,which is 4*TTX2. The value of this quantity is 4<lnrX2Vdt

To express, as is supposed, that the action of the envelopesupplies the place, at every instant, of that which would result fromthe presence of the medium, it is sufficient to equate the quantity

(11)

4th7rX2Vdt to the value which the expression — iKwX2-,-dt

receives when we give to x its complete value X ; hence we obtain

the equation ^ = - -£v9 which must hold when in the functions(XX £\-

— and v we put instead of x its value X, which we shall denotedx r

by writing it in the form K -r- +k V= 0.

116. The value of -^- taken when x = X, must therefore havedx

a constant ratio — ^ to the value of v, which corresponds to the

same point. Thus we shall suppose that the external cause ofthe cooling determines always the state of the very thin envelope,

r/i)in such a manner that the value of -j- which results from this

dx

state, is proportional to the value of v, corresponding to x = X}

and that the constant ratio of these two quantities is — -^ . This

condition being fulfilled by means of some cause always present,

which prevents the extreme value of -j- from being anything elsedx

but — ~F v, the action of the envelope will take the place of thatof the air.

It is not necessary to suppose the envelope to be extremelythin, and it will be seen in the sequel that it may have anindefinite thickness. Here the thickness is considered to beindefinitely small, so as to fix the attention on the state of thesurface only of the solid.

117. Hence it follows that the three equations which arerequired to determine the function </> (x, t) or v are the following,

dv K (d\ 2 dv\

[ + )The first applies to all possible values of x and t; the second

is satisfied when x = X} whatever be the value of t; and thethird is satisfied when t = 0, whatever be the value of x.

SECT. III.] VARIED MOVEMENT IN A CYLINDER. 95

I t might be supposed that in the initial state all the sphericallayers have not the same temperature : which is what wouldnecessarily happen, if the immersion were imagined not to havelasted for an indefinite time. In this case, which is more generalthan the foregoing, the given function, which expresses theinitial temperature of the molecules situated at distance x fromthe centre of the sphere, will be represented by F (x) ; the thirdequation will then be replaced by the following, </> {x, 0) =F(x).

Nothing more remains than a purely analytical problem,whose solution will be given in one of the following chapters.It consists in finding the value of v, by means of the generalcondition, and the two special conditions to which it is subject.

SECTION III.

Equations of the varied movement of heat in a solid cylinder.

118. A solid cylinder of infinite length, whose side is per-pendicular to its circular base, having been wholly immersedin a liquid whose temperature is uniform, has been graduallyheated, in such a manner that all points equally distant fromthe axis have acquired the same temperature; it is then exposedto a current of colder air; it is required to determine thetemperatures of the different layers, after a given time.

x denotes the radius of a cylindrical surface, all of whosepoints are equally distant from the axis ; X is the radius ofthe cylinder ; v is the temperature which points of the solid,situated at distance x from the axis, must have after the lapseof a time denoted by ty since the beginning of the cooling.Thus v is a function of x and t, and if in it t be made equal to0, the function of x which arises from this must necessarily satisfythe initial state, which is arbitrary.

119. Consider the movement of heat in an infinitely thinportion of the cylinder, included between the surface whoseradius is x, and that whose radius is x + dx. The quantity ofheat which this portion receives during the instant dt, from thepart of the solid which it envelops, that is to say, the quantitywhich during the same time crosses the cylindrical surface


whose radius is x, and whose length is supposed to be equalto unity, is expressed by

dx

To find the quantity of heat which, crossing the second surfacewhose radius is x + dx, passes from the infinitely thin shell intothe part of the solid which envelops it, we must, in the foregoingexpression, change x into x-\-dx, or, which is the same thing,add to the term

— 2KlTX -y- dt,dx

the differential of this term, taken with respect to x. Hencethe difference of the heat received and the heat lost, or thequantity of heat which accumulating in the infinitely thin shelldetermines the changes of temperature, is the same differentialtaken with the opposite sign, or

on the other hand, the volume of this intervening shell is 2irxdx}

and 2GDirxdx expresses the quantity of heat required to raiseit from the temperature 0 to the temperature 1, C being thespecific heat, and D the density. Hence the quotient

2Kir.dt.d x^V dx,

2CDirxdxis the increment which the temperature receives during theinstant dt. Whence we obtain the equation

dfc _ K^ (d2v 1 dv\dt CD \dx2 x dx) '

120. The quantity of heat which, during the instant dt,crosses the cylindrical surface whose radius is x, being expressed

dvin general by 2K7rx-^dt} we shall find that quantity which

escapes during the same time from the surface of the solid, bymaking x = X in the foregoing value; on the other hand/the

SECT. IV.] STEADY MOVEMENT IN A PRISM. 97

same quantity, dispersed into the air, is, by the principle of thecommunication of heat, equal to 2irXhvdt) we must therefore

have at the surface the definite equation —K-j~=hv. The

nature of these equations is explained at greater length, eitherin the articles which refer to the sphere, or in those wherein thegeneral equations have been given for a body of any form what-ever. The function v which represents the movement of heat inan infinite cylinder must therefore satisfy, 1st, the general equa-,. dv K fd2v 1 dv\ , . . v . ,tion — = -~ = -y-r, -\ T- which applies whatever x and t may

dt CD \dx" x dx) ' r r J

be; 2nd, the definite equation -^ v -f -7- = 0, which is true, whateverthe variable t may be, when x = X; 3rd, the definite equationv = F(x). The last condition must be satisfied by all valuesof v, when t is made equal to 0, whatever the variable x maybe. The arbitrary function F(x) is supposed to be known ; itcorresponds to the initial state.

SECTION IV.

Equations of the uniform movement of heat in a solid prismof infinite length

121. A prismatic bar is immersed at one extremity in aconstant source of heat which maintains that extremity at thetemperature A ; the rest of the bar, whose length is infinite,continues to be exposed to a uniform current of atmospheric airmaintained at temperature 0; it is required to determine thehighest temperature which a given point of the bar can' acquire.

The problem differs from that of Article 73, since we nowtake into consideration all the dimensions of the solid, which isnecessary in order to obtain an exact solution.

We are led, indeed, to suppose that in a bar of very smallthickness all points of the same section would acquire sensiblyequal temperatures; but some uncertainty may rest on theresults of this hypothesis. It is therefore preferable to solve theproblem rigorously, and then to examine, by analysis, up to whatpoint, and in what cases, we are justified in considering thetemperatures of different points of the same section to be equal.

F. H. 7

98 THEORY OF HEAT. [CHAP. IL

122. The section made at right angles to the length of thebar, is a square whose side is 2/, the axis of the bar is the axisof x, and the origin is at the extremity A. The three rectangularco-ordinates of a point of the bar are x, y} z} and v denotes thefixed temperature at the same point.

The problem consists in determining the temperatures whichmust be assigned to different points of the bar, in order thatthey may continue to exist without any change, so long as theextreme surface A, which communicates with the source of heat,remains subject, at all its points, to the permanent tempera-ture A ; thus v is a function of xy y, and z.

123. Consider the movement of heat in a prismatic molecule,enclosed between six planes perpendicular to the three axesof x, y, and z. The first three planes pass through the point mwhose co-ordinates are x, y, zy and the others pass through thepoint m whose co-ordinates are x + dx, y + dy> z + dz.

To find what quantity of heat enters the molecule duringunit of time across the first plane passing through the point mand perpendicular to x, we must remember that the extent of thesurface of the molecule on this plane is dydz, and that the flowacross this area is, according to the theorem of Article 98, equal

to —K—-\ thus the molecule receives across the rectangle dydz

passing through the point m a quantity of heat expressed by

— Kdydz -y-. To find the quantity of heat which crosses the

opposite face, and escapes from the molecule, we must substitute,in the preceding expression, x + dx for x, or, which is the samething, add to this expression its differential taken with respectto x only; whence we conclude that the molecule loses, at itssecond face perpendicular to x, a quantity of heat equal to

we must therefore subtract this from that which enters at theopposite face; the differences of these two quantities is

Kdydzd^j, or,

SECT. IV.] STEADY MOVEMENT IN A PEISM. 99

this expresses the quantity of heat accumulated in the moleculein consequence of the propagation in direction of x; which ac-cumulated heat would make the temperature of the moleculevary, if it were not balanced by that which is lost in some otherdirection.

It is found in the same manner that a quantity of heat equal

to —Kdzdx-r- enters the molecule across the plane passing

through the point m perpendicular to y, and that the quantitywhich escapes at the opposite face is

— Kdzdx^ Kdzdccd (4-) >dy \dy)

the last differential being taken with respect to y only. Hence

the difference of the two quantities, or Kdxdydz j - ^ y expressesay

the quantity of heat which the molecule acquires, in consequenceof the propagation in direction of y.

Lastly, it is proved in the same manner that the moleculeacquires, in consequence of the propagation in direction of z,a quantity of heat equal to Kdxdydz-p^ Now, in order that

there may be no change of temperature, it is necessary for themolecule to retain as much heat as it contained at first, so thatthe heat it acquires in one direction must balance that which itloses in another. Hence the sum of the three quantities of heatacquired must be nothing; thus we form the equation

d2v d2vdx dy dz

dv dv _dy d

124. It remains now to express the conditions relative to thesurface. If we suppose the point m to belong to one of the facesof the prismatic bar, and the face to be perpendicular to z, wesee that the rectangle dxdy, during unit of time, permits aquantity of heat equal to Vh dx dy to escape into the air,V denoting the temperature of the point m of the surface, namelywhat <j> (x, y, z) the function sought becomes when z is madeequal to I, half the dimension of the prism. On the other hand,the quantity of heat which, by virtue of the action of the

7—2

molecules, during unit of time, traverses an infinitely small surfaceto, situated within the prism, perpendicular to z, is equal to

— Kco-^, according to the theorems quoted above. This ex-dz

pression is general, and applying it to points for which the co-ordinate z has its complete value I, we conclude from it that thequantity of heat which traverses the rectangle dxdy taken at the

surface h -Kdxdy-^, giving to z in the function ^ its com-

dvplete value I Hence the two quantities —Kdxdy-j-, and

h dx dy v, must be equal, in order that the action of the moleculesmay agree with that of the medium. This equality must also

dvexist when we give to z in the functions -j- and v the value — I,

dzwhich it has at the face opposite to that first considered. Further,the quantity of heat which crosses an infinitely small surface w}

dvperpendicular to the axis of y, being —Kco-j-. it follows that

aythat which flows across a rectangle dz dx taken on a face of the

dvprism perpendicular to y is - K dzdx-j- , giving to y in the

ay

function -=- its complete value I. Now this rectangle dz doc

permits a quantity of heat expressed by hv dx dy to escape into

the air; the equation hv = — K^ becomes therefore necessary,when v is made equal to I or — I in the functions v and -^-.

dy125. The value of the function v must by hypothesis be

equal to A, when we suppose x = 0, whatever be the values ofy and z. Thus the required function v is determined by thefollowing conditions: 1st, for all values of x, y, z} it satisfies thegeneral equation

<Fv dh d2v _dx2 + d2,2 + ~d?-°>

2nd, it satisfies the equation ^ v + -^=0, when y is equal to

SECT. V.] VARIED MOVEMENT IN A' CUBE. 101

t or — I, whatever x and z may be, or satisfies the equation

-p-v + -j- = 0, when z is equal to I or — I, whatever x and y may-

be ; 3rd, it satisfies the equation v = A, when x = 0, whatever

y and z may be.

SECTION V.

Equations of the varied movement of heat in a solid cube.

126. A solid in the form of a cube, all of whose points haveacquired the same temperature, is placed in a uniform current ofatmospheric air, maintained at temperature 0. It is required todetermine the successive states of the body during the wholetime of the cooling.

The centre of the cube is taken as the origin of rectangularcoordinates; the three perpendiculars dropped from this point onthe faces, are the axes of x, y, and z \ 21 is the side of the cube,v is the temperature to which a point whose coordinates arex, y, zy is lowered after the time t has elapsed since the com-mencement of the cooling: the problem consists in determiningthe function v, which depends on x> y, z and t.

127. To form the general equation which v must satisfy,we must ascertain what change of temperature an infinitelysmall portion of the solid must experience during the instantdty by virtue of the action of the molecules which are extremelynear to it. We consider then a prismatic molecule enclosedbetween six planes at right angles; the first three pass throughthe point m, whose co-ordinates are x, y> z, and the three others,through the point m'y whose co-ordinates, are

x + dx, y + dyy z + dz.

The quantity of heat which during the instant dt passes intothe molecule across the first rectangle dydz perpendicular to x,

is —Kdydz-T- dt, and that which escapes in the same time from

the molecule, through the opposite face, is found by writingx + dx in place of x in the preceding expression, it is

- Kdy dz (~) dt - Kdy dz d Q dt,


the differential being taken with respect to x only. The quantityof heat which during the instant dt enters the molecule, acrossthe first rectangle dz dx perpendicular to the axis of y} is

d ??- Kdzdx-i-dt, and that which escapes from the molecule during

dythe same instant, by the opposite face, is

— Kdz dx^r- dt — Kdzdxd l-j-) dt,dy \dyj

the differential being taken with respect to y only. The quantityof heat which the molecule receives during the instant dt, through

dvits lower face, perpendicular to the axis of z, is — K dx dy -j- dt,

dz

and that, which it loses through the opposite face is

— Kdxdy-j-dt — Kdxdyd(-j-) dt,the differential being taken with respect to z only.

The sum of all the quantities of heat which escape from themolecule must now be deducted from the sum of the quantitieswhich it receives, and the difference is that which determines itsincrease of temperature during the instant: this difference is

Kdy dz d Q dt + Kdz dx d (j^J dt + Kdxdyd (Jj dt,

d*v d2v+

128. If the quantity which has just been found be divided bythat which is necessary to raise the molecule from the temperature0 to the temperature 1, the increase of temperature which iseffected during the instant dt will become known. Now, thelatter quantity is CD dx dy dz : for C denotes the capacity ofthe substance for heat; D its density, and dxdydz the volumeof the molecule. The movement of heat in the interior of thesolid is therefore expressed by the equation

di ~

SECT. V.] VARIED MOVEMENT IN A CUBE. 103

129. It remains to form the equations which relate to thestate of the surface, which presents no difficulty, in accordancewith the principles which we have established. In fact, thequantity of heat which, during the instant dt, crosses the -rectangle

dz dyy traced on a plane perpendicular to x> is — K dy dz -7- dt.

This result, which applies to all points of the solid, ought to hold

when the value of x is equal to I, half the thickness of the prism.

In this case, the rectangle dy dz being situated at the surface, the

quantity of heat which crosses it, and is dispersed into the air

during the instant dt, is expressed by hv dy dz dt, we ought there-

fore to have, when x = l, the equation hv = —K-j-. This con-

dition must also be satisfied when x = — I.It will be found also that, the quantity of heat which crosses

the rectangle dz dx situated on a plane perpendicular to the axisdv

of y being in general — Kdzdx-^, and that which escapes at the

surface into the air across the same rectangle being hvdzdxdt,

we must have the equation hv + K -y- = 0, when y = l or —I.dy

Lastly, we obtain in like manner the definite equation

dz

which is satisfied when z = I or — I.

130. The function sought, which expresses the varied move-ment of heat in the interior of a solid of cubic form, must thereforebe determined by the following conditions:

1st. It satisfies the general equation

dv_ K (d*v dSdt~ C.lJ\d2+ d*

2nd. It satisfies the three definite equations

hv + K-y- = 0, hv + K -j- = 0, hv+Kj- = 09dx dy dz

which hold when x = ± I, y = ± I, s = ±l;


3rd. If in the function v which contains x, y, z, t, we maket = 0, whatever be the values of x, y, and z, we ought to have,according to hypothesis, v = A, which is the initial and commonvalue of the temperature.

131. The equation arrived at in the preceding problemrepresents the movement of heat in the interior of all solids.Whatever, in fact, the form of the body may be, it is evident that,by decomposing it into prismatic molecules, we shall obtain thisresult. We may therefore limit ourselves to demonstrating inthis manner the equation of the propagation of heat. But inorder to make the exhibition of principles more complete, andthat we may collect into a small number of consecutive articlesthe theorems which serve to establish the general equation of thepropagation of heat in the interior of solids, and the equationswhich relate to the state of the surface, we shall proceed, in thetwo following sections, to the investigation of these equations,independently of any particular problem, and without revertingto the elementary propositions which we have explained in theintroduction.

SECTION VI.

General equation of the propagation of heat in the interior of solids.

132. THEOREM I. If the different points of a homogeneoussolid mass, enclosed between six planes at right angles, have actualtemperatures determined by the linear equation

v = A—ax -by — cz, (a),

and if the molecules situated at the external surface on the sixplanes ivhich bound the prism are maintained, by any cause what-ever, at the temperature expressed by the equation (a) : all themolecules situated in the interior of the mass will of themselvesretain their actual temperatures, so that there %vitt be no change inthe state of the prism.

v denotes the actual temperature of the point whose co-ordinates are x, y, z; A, a, b, c, are constant coefficients.

To prove this proposition, consider in the solid any threepoints whatever wJ/ju, situated on the same straight line

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 105

which the point M divides into two equal parts; denote byos, y, z the co-ordinates of the point M, and its temperature byv, the co-ordinates of the point /JU by x + a, y + fi, z + y, and itstemperature by w, the co-ordinates of the point m by x - a, y — /3,z — y, and its temperature by u, we shall have

v = A — ax — by — cz,

w =• A — a (x + a) - b (y + /3) - c (z + 7),

11 = A — a (x — a) — b (y — /3) — c (z — 7),

whence we conclude that,

v — w = ai + b/3 + cy, and u — v = ax + bfi + cy ;

therefore v — iv = u — v.

Now the quantity of heat which one point receives fromanother depends on the distance between the two points andon the difference of their temperatures. Hence the action ofthe point M on the point /-& is equal to the action of m on M;thus the point M receives as much heat from m as it gives upto the point /JL.

We obtain the same result, whatever be the direction andmagnitude of the line which passes through the point il/? andis divided into two equal parts. Hence it is impossible for thispoint to change its temperature, for it receives from all partsas much heat as it gives up.

The same reasoning applies to all other points; hence nochange can happen in the state of the solid.

133. COROLLARY I. A solid being enclosed between twoinfinite parallel planes A and B, if the actual temperature ofits different points is supposed to be expressed by the equationv = 1 — z, and the two planes which bound it are maintainedby any cause whatever, A at the temperature 1, and B at thetemperature 0; this particular case will then be included inthe preceding lemma, if we make A =1, a = 0, b = 0, c = 1.

134. COROLLARY II. If in the interior of the same solidwe imagine a plane M parallel to those which bound it, we seethat a certain quantity of heat flows across this plane duringunit of time ; for two very near points, such as m and n, one

106 THEORY OF HEAT. [CHAP. IT.

of which is below the plane and the other above it, are unequallyheated; the first, whose temperature is highest, must thereforesend to the second, during each instant, a certain quantity of heatwhich, in some cases, may be very small, and even insensible,according to the nature of the body and the distance of the twomolecules.

The same is true for any two other points whatever separatedby the plane. That which, is most heated sends to the othera certain quantity of heat, and the sum of these partial actions,or of all the quantities of heat sent across the plane, composesa continual flow whose value does not change, since all themolecules preserve their temperatures. It is easy to prove thatthis flow, or the quantity of heat which crosses the plane M duringthe unit of time, is equivalent to that ivhich crosses, during the sametime, another plane N parallel to the first. In fact, the part ofthe mass which is enclosed between the two surfaces M andN will receive continually, across the plane M, as much heatas it loses across the plane N. If the quantity of heat, whichin passing the plane M enters the part of the mass which isconsidered, were not equal to that which escapes by the oppositesurface JV, the solid enclosed between the two surfaces wouldacquire fresh heat, or would lose a part of that which it has,and its temperatures would not be constant; which is contrary tothe preceding lemma.

135. The measure of the specific conducibility of a givensubstance is taken to be the quantity of heat which, in an infinitesolid, formed of this substance, and enclosed between two parallelplanes, flows during unit of time across unit of surface, takenon any intermediate plane whatever, parallel to the externalplanes, the distance between which is equal to unit of length,one of them being maintained at temperature 1, and the otherat temperature 0. This constant flow of the heat which crossesthe whole extent of the prism is denoted by the coefficient K,and is the measure of the conducibility.

136. LEMMA. If we suppose all the temperatures of the solid inquestion under the preceding article, to be multiplied by any numberwhatever g, so that the equation of temperatures is v = g - g z ,instead of being v = 1 - z, and if the two external planes are main-


tained, one at the temperature g, and the other at temperature 0,the constant flow of heat, in this second hypothesis, or the quantityivhich during unit of time crosses unit of surface taken on anintermediate plane parallel to the bases, is equal to the productof the first flow multiplied by g.

In fact, since all the temperatures have been increased inthe ratio of 1 to g, the differences of the temperatures of anytwo points whatever m and fi, are increased in the same ratio.Hence, according to the principle of the communication of heat,in order to ascertain the quantity of heat which m sends to fion the second hypothesis, we must multiply by g the quantitywhich the same point m sends to ft on the first hypothesis.The same would be true for any two other points whatever.Now, the quantity of heat which crosses a plane M results fromthe sum of all the actions which the points m, m, m", m", etc.,situated on the same side of the plane, exert on the points /u,IM, fi", fi", etc., situated on the other side. Hence, if in the firsthypothesis the constant flow is denoted by K, it will be equal togK, when we have multiplied all the temperatures by g.

137. THEOREM II. In a prism whose constant temperaturesare expressed by the equation v = A — ax — by — cz, and whichis bounded by six planes at right angles all of whose points aremaintained at constant temperatures determined by the precedingequation, the quantity of heat which, during unit of time, crossesunit of surface taken on any intermediate plane tuhatever perpen-dicular to z, is the same as the constant flow in a solid of thesame substance would be, if enclosed between two infinite parallelplanes, and for which the equation of constant temperatures isv = c — cz.

To prove this, let us consider in the prism, and also in theinfinite solid, two extremely near points m and fi, separated

Fig. 4.

m

by the plane M perpendicular to the axis of z; fi being abovethe plane, and m below it (see fig. 4), and above the same plane

108 THEOKY OF HEAT. [CHAP. II.

let us take a point m such that the perpendicular dropped fromthe point /UL on the plane may also be perpendicular to thedistance mm at its middle point h. Denote by x, y, z + h} theco-ordinates of the point JJL, whose temperature is w, by x - a, y - &z, the co-ordinates of m, whose temperature is v, and by x + a,y + /3, z, the co-ordinates of m, whose temperature is v.

The action of m on fi, or the quantity of heat which m sendsto /n during a certain time, may be expressed by q (v — w). Thefactor q depends on the distance m/jb, and on the nature of themass. The action of m' on /i will therefore be expressed byq (v — w); and the factor q is the same as in the precedingexpression; hence the sum of the two actions of m on fi, andof m on jjb, or the quantity of heat which JJL receives from m andfrom m, is expressed by

q (v — w -\- v —w).

Now, if the points m9 p, m belong to the prism, we have

w — A — ax — by - c (z + h), v = A — a (x — a) — b (y — /3) - cz,

and v = A — a (x + a) — b (y + /3) — cz ;

and if the same points belonged to an infinite solid, we shouldhave, by hypothesis,

w = c — c [z + h), v = c — cz, and v = c — cz.

In the first case, we find

q (v — w + v — w) = 2qchy

and, in the second case, we still have the same result. Hencethe quantity of heat which JA receives from m and from m onthe first hypothesis, when the equation of constant temperaturesis v = A-ax — by — cz, is equivalent to the quantity of heatwhich fi receives from m and from m when the equation ofconstant temperatures is v = c — cz.

The same conclusion might be drawn with respect to any threeother points whatever m', /*', m\ provided that the second y! beplaced at equal distances from the other two, and the altitude ofthe isosceles triangle m / / m be parallel to z. Now, the quantityof heat which crosses any plane whatever M, results from the sumof the actions which all the points m, m', m\ m" etc., situated on


one side of this plane, exert on all the points /JL} //, fi"9 /JL", etcsituated on the other side: hence the constant flow, which, duringunit of time, crosses a definite part of the plane M in the infinitesolid, is equal to the quantity of heat which flows in the same timeacross the same portion of the plane M in the prism, all of whosetemperatures are expressed by the equation

v = A—ax — by -cz.

138. COROLLARY. The flow has the value cK in the infinitesolid, when the part of the plane which it crosses has unit of

surface. In the prism also it has the same value cK or — K j - .

It is proved in the same manner, that the constant flow which takesplace, during unit of time, in the same prism across unit of surface,on any plane whatever 'perpendicular to y, is equal to

bK or - K ^ ;

and that ivhich crosses a plane perpendicidar to x has the value

dva K o r - K - r - .

dx

139. The propositions which we have proved in the precedingarticles apply also to the case in which the instantaneous action ofa molecule is exerted in the interior of the mass up to an appre-ciable distance. In this case, we must suppose that the causewhich maintains the external layers of the body in the stateexpressed by the linear equation, affects the mass up to a finitedepth. All observation concurs to prove that in solids and liquidsthe distance in question is extremely small.

140. THEOREM III. If the temperatures at the points of asolid are expressed by the equation v = f (x} y, z, t), in whichx, y> z are the co-ordinates of a molecule whose temperature isequal to v after the lapse of a time t; the flow of heat whichcrosses part of a plane traced in the solid, perpendicular to one ofthe three axes, is no longer constant; its value is different fordifferent parts of the plane, and it varies also with the time. Thisvariable quantity may be determined by analysis.

H O THEORY OF HEAT. [CHAP. II.

Let a) be an infinitely small circle whose centre coincides withthe point m of the solid, and whose plane is perpendicular to thevertical co-ordinate z ; during the instant dt there will flow acrossthis circle a certain quantity of heat which will pass from thepart of the circle below the plane of the circle into the upperpart. This flow is composed of all the rays of heat which departfrom a lower point and arrive at an upper point, by crossinga point of the small surface w. We proceed to shew that the

expression of the value of the flow is — K j - o>dt.

Let us denote by x, y\ z the coordinates of the point m whosetemperature is v ; and suppose all the other molecules to bereferred to this point m chosen as the origin of new axes parallelto the former axes : let £ 77, £ be the three co-ordinates of a pointreferred to the origin m; in order to express the actual temperaturew of a molecule infinitely near to m, we shall have the linearequation

, ^ dv dv , dvb dx dy dz

The coefficients v\ -7—, -y-, -r- are the values which are foundax ay dz

by substituting in the functions v, —r > --,- , -7-, for the variablesJ & dx dt/ dzx, y z, the constant quantities x, y\ z, which measure the dis-tances of the point m from the first three axes of x, y, and z.

Suppose now that the point in is also an internal molecule ofa rectangular prism, enclosed between six planes perpendicular tothe three axes whose origin is m; that w the actual temperature ofeach molecule of this prism, whose dimensions are finite, is ex-pressed by the linear equation w = A + af; + brj + c£ and that thesix faces which bound the prism are maintained at the fixed tem-peratures which the last equation assigns to them. The state ofthe internal molecules will also be permanent, and a quantity ofheat measured by the expression —Kcivdt will flow during theinstant dt across the circle co.

This arranged, if we take as the values of the constantsA 7 xi J.'.- > dv dv dv' .

A, a, 6, c, the quantities v, j ^ y ^ - , -j-, the fixed state of the

SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. I l l

prism will be expressed by the equation

, , dv ^ dv dv „dx* dy ' dz

Thus the molecules infinitely near to the point m will have,during the instant dty the same actual temperature in the solidwhose state is variable, and in the prism whose state is constant.Hence the flow which exists at the point m, during the instant dt,across the infinitely small circle co, is the same in either solid; it

is therefore expressed by — K -j— eodt.

From this we derive the following propositionIf in a solid whose internal temperatures vary with the time, by

virtue of the action of the molecules, we trace any straight line what-ever, and erect (see fig. 5), at the different points of this line, theordinates pm of a plane curve equal to the temperatures of thesepoints taken at the same moment; the floiv of heat, at each point pof the straight line, will be proportional to the tangent of the anglea which the element of the curve makes ivith the parallel to theabscisses; that is to say, if at the point p we place the centre of an

Fig. 5.

infinitely small circle w perpendicular to the line, the quantity ofheat which has flowed during the instant dt, across this circle, inthe direction in which the abscissae op increase, will be measuredby the product of four factors, which are, the tangent of the anglea, a constant coefficient K, the area co of the circle, and the dura-tion dt of the instant.

141. COROLLARY. If we represent by e the abscissa of thiscurve or the distance of a point y of the straight line from a

112 THEOKY OF HEAT. [CHAP. II.

fixed point o, and by v the ordinate which represents the tem-perature of the point p, v will vary with the distance e andwill be a certain function f(e) of that distance; the quantityof heat which would flow across the circle co, placed at the

fii)point p perpendicular to the line, will be — K -j- oodt, or

-Kf'{e)a)dt,

denoting the function ^ by f{e).

We may express this result in the following manner, whichfacilitates its application.

To obtain the actual floiv of heat at a point p of a straightline drawn in a solid, whose temperatures vary by action of themolecules9 we must divide the difference of the temperatures attwo points infinitely near to the point p by the distance betweenthese points. The flow is proportional to the quotient

142. THEOEEM IV. From the preceding Theorems it iseasy to deduce the general equations of the propagation of heat.

Suppose the different points of a homogeneous solid of anyform whatever, to have received initial temperatures which varysuccessively by the effect of the mutual action of the molecules,and suppose the equation v = f (x, y, z, t) to represent the successivestates of the solid, it may noiv be shewn that v a function of fourvariables necessarily satisfies the equation

dv _ K_ /d2v d2v dd ~ l V b ? + d7 + d

In fact, let us consider the movement of heat in a moleculeenclosed between six planes at right angles to the axes of x, y,and z; the first three of these planes pass through the pointm whose coordinates are x, y, z, the other three pass throughthe point m, whose coordinates are x + dx, y + dy, z + dz.

During the instant dt, the molecule receives, across thelower rectangle dxdy, which passes through the point m, a

quantity of heat equal to - K dx dy ~ dt To obtain the quantity

which escapes from the molecule by the opposite face, it issufficient to change z into z + dz in the preceding expression,


that is to say, to add to this expression its own differential takenwith respect to z only; we then have

4-)— Kdx du - j - dt —Kdx dy \ dz dtJ dz ^ dz

as the value of the quantity which escapes across the upperrectangle. The same molecule receives also across the firstrectangle dz dx which passes through the point m, a quantity

(ID

of heat equal to — K~r~ dz dx dt; and if we add to this ex-dy

pression its own differential taken with respect to y only, wefind that the quantity which escapes across the opposite facedz dx is expressed by

— K^r- dz dxdt — K 1 J dy dz dx dt.

dy dyLastly, the molecule receives through the first rectangle dy dz

Cl 1)a quantity of heat equal to — K -j- dy dz dt, and that which it

CLX

loses across the opposite rectangle which passes through m isexpressed by

7 (dv\7 ^ T~

T^ClV ? , , „ \dXl 7 7 7 7 j l— K^r ay azat—K , ax du dz dt.dx u dx J

We must now take the sum of the quantities of heat whichthe molecule receives and subtract from it the sum of thosewhich it loses. Hence it appears that during the instant dty

a total quantity of heat equal to72 72 \

v a v a v\

accumulates in the interior of the molecule. It remains onlyto obtain the increase of temperature which must result fromthis addition of heat.

D being the density of the solid, or the weight of unit ofvolume, and G the specific capacity, or the quantity of heatwhich raises the unit of weight from the temperature 0 to thetemperature 1 ; the product CDdocdydz expresses the quantity

F. H. 8

of heat required to raise from 0 to 1 the molecule whose volumeis dxdydz. Hence dividing by this product the quantity ofheat which the molecule has just acquired, we shall have itsincrease of temperature. Thus we obtain the general equation

dSdv __ K fd2v <Fv <FvJt~cnW + df + d^

which is the equation of the propagation of heat in the interiorof all solid bodies.

143. Independently of this equation the system of tempera-tures is often subject to several definite conditions, of which nogeneral expression can be given, since they depend on the natureof the problem.

If the dimensions of the mass in which heat is propagated arefinite, and if the surface is maintained by some special cause in agiven state; for example, if all its points retain, by virtue of thatcause, the constant temperature 0, we shall have, denoting theunknown function v by <£ (x, y, z, t), the equation of condition<f> (x, y, z, t) = 0; which must be satisfied by all values of as, y, zwhich belong to points of the external surface, whatever be thevalue of t Further, if we suppose the initial temperatures of thebody to be expressed by the known function F(x, y, z), we havealso the equation <f> (x, y, z> 0) = F(x, y, z) ; the condition ex-pressed by this equation must be fulfilled by all values of theco-ordinates x, y, z which belong to any point whatever of thesolid.

144. Instead of submitting the surface of the body to a con-stant temperature, we may suppose the temperature not to bethe same at different points of the surface, and that it varies withthe time according to a given law; which is what takes place inthe problem of terrestrial temperature. In this case the equationrelative to the surface contains the variable t

145. In order to examine by itself, and from a very generalpoint of view, the problem of the propagation of heat, the solidwhose initial state is given must be supposed to have all itsdimensions infinite; no special condition disturbs then the dif-

SECT. VII.] GENERAL SURFACE EQUATION. 115

fusion of heat, and the law to which this principle is submittedbecomes more manifest; it is expressed by the general equation

dv K /d2v , d2v dv2\

dt CD \dx2 ' dy2 ^ dz2j '

to which must be added that which relates to the initial arbitrarystate of the solid.

Suppose the initial temperature of a molecule, whose co-ordinates are xy y, z, to be a known function F{x, y, z), and denotethe unknown value v by <f> (oc, y, z, t), we shall have the definiteequation <£ (pc, y, z, 0) = F (x, y, z); thus the problem is reduced tothe integration of the general equation (A) in such a manner thatit may agree, when the time is zero, with the equation which con-tains the arbitrary function F.

SECTION VII.

General equation relative to the surface.

146. If the solid has a definite form, and if its original heatis dispersed gradually into atmospheric air maintained at a con-stant temperature, a third condition relative to the state of thesurface must be added to the general equation (A) and to thatwhich represents the initial state.

We proceed to examine, in the following articles, the nature ofthe equation which expresses this third condition.

Consider the variable state of a solid whose heat is dispersedinto air, maintained at the fixed temperature 0. Let co be aninfinitely small part of the external surface, and /JL a point of co,through which a normal to the surface is drawn; different pointsof this line have at the same instant different temperatures.

Let v be the actual temperature of the point /i, taken at adefinite instant, and w the corresponding temperature of a point vof the solid taken on the normal, and distant from \JU by an in-finitely small quantity a. Denote by x> y} z the co-ordinates ofthe point /-&, and those of the point v by x + S#? y + %, z + Sz ;l e ty (x9 y, #) = 0 be the known equation to the surface of the solid,and v = <j) (x, yy z, t) the general equation which ought to give the

8—2

116 THEORY OF HEAT. [CHAP. IT.

value of v as a function of the four variables x, y, z, t. Differen-tiating the equation f(x, y, z) = 0, we shall have

mdoc + ndy +pdz = 0 ;

m, n, p being functions of oo} y, z.It follows from the corollary enunciated in Article 141, that

the flow in direction of the normal, or the quantity of heat whichduring the instant dt would cross the surface a>, if it were placedat any point whatever of this line, at right angles to its direction,is proportional to the quotient which is obtained by dividing thedifference of temperature of two points infinitely near by theirdistance. Hence the expression for the flow at the end of thenormal is

— A coat;a

K denoting the specific conducibility of the mass. On the otherhand, the surface co permits a quantity of heat to escape into theair, during the time dt, equal to hvcodt; h being the conducibilityrelative to atmospheric air. Thus the flow of heat at the end ofthe normal has two different expressions, that is to say :

hvcodt and - K —— codt:a

hence these two quantities are equal; and it is by the expressionof this equality that the condition relative to the surface is in-troduced into the analysis.

147. We have

dx dy y dz Z*

Now, it follows from the principles of geometry, that the co-ordinates 8x, 8yy 8z, which fix the position of the point v of thenormal relative to the point p, satisfy the following conditions:

p8x = m8z,

We have therefore

1 / dv dvw — v = - " '

SECT. VII.J GENERAL SURFACE EQUATION. H7

we have also

a = JSa?+8if + $s? = - (m2 + n2 + p*)h Bz9

or a = ^ Bz, denoting by 5 the quantity (m2 + n2 + p2)*,

1 w — v / dv dv , cfaA 1hence = m -7— + w -y- 4- »-^- - ;

a \ rfa; dy l dzj q

consequently the equation

hvcodt = - k [ ) codt\ a J

becomes the following1 :dv dv dv h A /RN

ax dy * dx K *This equation is definite and applies only to points at the

surface ; it is that which must be added to the general equation ofthe propagation of heat (A), and to the condition which deter-mines the initial state of the solid; m, n, p, q, are known functionsof the co-ordinates of the points on the surface.

148. The equation (B) signifies in general that the decrease ofthe temperature, in the direction of the normal, at the boundary ofthe solid, is such that the quantity of heat which tends to escapeby virtue of the action of the molecules, is equivalent always tothat which the body must lose in the medium.

The mass of the solid might be imagined to be prolonged,in such a manner that the surface, instead of being exposed to theair, belonged at the same time to the body which it bounds, andto the mass of a solid envelope which contained it. If, on thishypothesis, any cause whatever regulated at every instant thedecrease of the temperatures in the solid envelope, and determinedit in such a manner that the condition expressed by the equation(B) was always satisfied, the action of the envelope would take the

1 Let N be the normal,

the rest as in the text. [R. L. E.]

dN

dv vi dv

118 THEORY OF HEAT. |_CHAP. II.

place of that of the air, and the movement of heat would be thesame in either case : we can suppose then that this cause exists,and determine on this hypothesis the variable state of the solid;which is what is done in the employment of the two equations(A) and (B).

By this it is seen how the interruption of the mass and theaction of the medium, disturb the diffusion of heat by submittingit to an accidental condition.

149. We may also consider the equation (B), which relatesto the state of the surface under another point of view : but wemust first derive a remarkable consequence from Theorem in.(Art. 140). We retain the construction referred to in the corollaryof the same theorem (Art. 141). Let x, y, z be the co-ordinatesof the point p, and

x + $x, y + $y, z + Bz

those of a point q infinitely near to py and taken on the straightline in question: if we denote by v and iv the temperatures of thetwo points p and q taken at the same instant, we have

, rs , dv ~ dv * dv <>w = v + ov = v + - j - ox + -j~ by + -j- oz ;

(X/Oo ^ y ^^hence the quotient

Sv dv Sx dv dy dv Sz

thus the quantity of heat which flows across the surface ay placedat the point m, perpendicular to the straight line, is

dv Bx dv Sy dv S

The first term is the product of - K~ by dt and b v w -dx J J he

The latter quantity is, according to the principles of geometry, thearea of the projection of co on the plane of y and z ; thus theproduct represents the quantity of heat which would flow acrossthe area of the projection, if it were placed at the point p perpen-dicular to the axis of x.


The second term - K-j- co-^-dt represents the quantity of

heat which would cross the projection of &>, made on the plane ofx and z} if this projection were placed parallel to itself at thepoint p.

Lastly, the third term — K-^-co j-dt represents the quantity

of heat which would flow during the instant dt, across the projec-tion of co on the plane of x and y, if this projection were placed atthe point p, perpendicular to the co-ordinate z.

By this it is seen that the quantity of heat which flows acrossevery infinitely small part of a surface drawn in the interior of thesolid, can akvays be decomposed into three other quantities of flow,which penetrate the three orthogonal projections of the surface, alongthe directions perpendicular to the planes of the projections. Theresult gives rise to properties analogous to those which havebeen noticed in the theory of forces.

150. The quantity of heat which flows across a plane surfaceo), infinitely small, given in form and position, being equivalentto that which would cross its three orthogonal projections, it fol-lows that, if in the interior of the solid an element be imagined ofany form whatever, the quantities of heat which pass into thispolyhedron by its different faces, compensate each other recipro-cally: or more exactly, the sum of the terms of the first order,which enter into the expression of the quantities of heat receivedby the molecule, is zero ; so that the heat which is in fact accumu-lated in it, and makes its temperature vary, cannot be expressedexcept by terms infinitely smaller than those of the first order.

This result is distinctly seen when the general equation (A)has been established, by considering the .movement of heat ina prismatic molecule (Articles 127 and 142); the demonstrationmay be extended to a molecule of any form whatever, by sub-stituting for the heat received through each face, that which itsthree projections would receive.

In other respects it is necessary that this should be so : for, ifone of the molecules of the solid acquired during each instant aquantity of heat expressed by a term of the first order, the varia-tion of its temperature would be infinitely greater than that of


other molecules, that is to say, during each infinitely small instantits temperature would increase or decrease by a finite quantity,which is contrary to experience.

151. We proceed to apply this remark to a molecule situatedat the external surface of the solid.

Fig. 6.

a

a

h

c

Through a point a (see fig. 6), taken on the plane of x and y,draw two planes perpendicular, one to the axis of x the other tothe axis of y. Through a point b of the same plane, infinitelynear to a, draw two other planes parallel to the two precedingplanes; the ordinates z, raised at the points a, b, c, d, up to theexternal surface of the solid, will mark on this surface four pointsa, b\ c', d\ and will be the edges of a truncated prism, whose baseis the rectangle abed. If through the point a which denotes theleast elevated of the four points a, b\ c, d', a plane be drawnparallel to that of x and y, it will cut off from the truncated prisma molecule, one of whose faces, that is to say a'b'cd', coincideswith the surface of the solid. The values of the four ordinatesaa\ cc, dd', bbf are the following:

aa =

f dz 7cc = z -h -Y~ ax,dx '

ou = z + -7- dx + -,—dx dy


152. One of the faces perpendicular to x is a triangle, andthe opposite face is a trapezium. The area of the triangle is

1 , dz ,2*/^*/'

and the flow of heat in the direction perpendicular to this surfacedv

being — K -7- we have, omitting the factor dt,CLOG

Trdv 1 7 dz 7K d d

as the expression of the quantity of heat which in one instantpasses into the molecule, across the triangle in question.

The area of the opposite face is

0 , dz ., dz

and the flow perpendicular to this face is also — K-^-y suppress-ing terms of the second order infinitely smaller than those of thefirst; subtracting the quantity of heat which escapes by the secondface from that which enters by the first we find

^ dv dz

This term expresses the quantity of heat the molecule receivesthrough the faces perpendicular to x.

It will be found, by a similar process, that the same moleculereceives, through the faces perpendicular to y, a quantity of heat

, , Tr dv dz , 7

equal to K -= 7— ax dy.^ dy dy u

The quantity of heat which the molecule receives through the

rectangular base is — K-j-dx dy. Lastly, across the upper sur-

face a'b'cd'y a certain quantity of heat is permitted to escape,

equal to the product of hv into the extent co of that surface.

The value of co is, according to known principles, the same as thatof dx dy multiplied by the ratio - ; e denoting the length of the

znormal between the external surface and the plane of x and yy and

'dz\2 /dz\2

\dx) \dy)


hence the molecule loses across its surface db'c'd' a quantity of

heat equal to hv dx dy -.z

Now, the terms of the first order which enter into the expressionof the total quantity of heat acquired by the molecule, must canceleach other, in order that the variation of temperature may not beat each instant a finite quantity; we must then have the equation

h e dv dz dv dz dvK z" dx dx dy dy dz'

153. Substituting for -£- and -j- their values derived from

the equation

mdx + ndy +pdz = 0,

and denoting by q the quantity

(m2+n2+p2) ,we have

thus we know distinctly what is represented by each of theterms of this equation.

Taking them all with contrary signs and multiplying themby dx dy, the first expresses how much heat the molecule receivesthrough the two faces perpendicular to x, the second how muchit receives through its two faces perpendicular to y, the thirdhow much it receives through the face perpendicular to z, andthe fourth how much it receives from the medium. The equationtherefore expresses that the sum of all the terms of the firstorder is zero, and that the heat acquired cannot be representedexcept by terms of the second order.

154. To arrive at equation (B), we in fact consider oneof the molecules whose base is in the surface of the solid, asa vessel which receives or loses heat through its different faces.The equation signifies that all the terms of the first order which

SECT. VIII.J GENERAL EQUATIONS APPLIED. 123

enter into the expression of the heat' acquired cancel each other;so that the gain of heat cannot be expressed except by termsof the second order. We may give to the molecule the form,either of a right prism whose axis is normal to the surface of thesolid, or that of a truncated prism, or any form whatever.

The general equation (A), (Art. 142) supposes that all theterms of the first order cancel each other in the interior of themass, which is evident for prismatic molecules enclosed in thesolid. The equation (B), (Art. 147) expresses the same resultfor molecules situated at the boundaries of bodies.

Such are the general points of view -from which we may lookat this part of the theory of heat.

mi ,. dv K (d2v d2v d*v\ , ,,I he equation -j- = 7T=^ -y- + - - + —„ represents the move-

^ dt CD \dx2 dy2 dz2/ r

ment of heat in the interior of bodies. It enables us to ascer-tain the distribution from instant to instant in all substancessolid or liquid; from it we may derive the equation whichbelongs to each particular case.

In the two following articles we shall make this applicationto the problem of the cylinder, and to that of the sphere.

SECTION VIII.

Application of the general equations.

155. Let us denote the variable radius of any cylindricalenvelope by r, and suppose, as formerly, in Article 118, thatall the molecules equally distant from the axis have at eachinstant a common temperature; JV will be a function of r and t\r is a function of y, z, given by the equation r2 = y2 + z2. It isevident in the first place that the variation of v with respect

d2vto x is nul; thus the term -^ must be omitted. We shall have

then, according to the principles of the differential calculus, theequations

dv _dv dr , d2v d2v /dr\2 dv /d2rdy = ~drdy ~d/ = d? \dy) + ~dr [dy\

dv_dvdr crv_(Tv_fdr\2 dv fd?r\Tz"TrTz dz2 " dr1 \dz) + dr WJ >

124 THEOEY OF HEAT. [CHAP. II.

whence

d2v d\ d*v (/dry , (dr\2 , dv fd*r drr\dJ + d?-d?\W +{lJ + dW + d*)

In the second member of the equation, the quantities

dr dr d2r d2rdy ' dz y dy2' dz2 '

must be replaced by their respective values; for which purposewe derive from the equation y2 + z2 = r2,

dr , - fdr\2 d2ry = r -7- and 1 = -7- 1 + r -j—2 ,u dy \dyj dy

dr . 1 (dr\2 d2rd 1 [ I +

dr (dry tz^r-r and 1 = [ — I + r-^^ ,dz \dzj dzand consequently

The first equation, whose first member is equal to r2, gives

(dr\2 ,

the second gives, when we substitute for

(drY (dry\dy) + [dz)

its value 1,

If the values given by equations (b) and (c) be now substi-tuted in (a), we have

d?v dh_dh ldv

Hence the equation which expresses the movement of heatin the cylinder, is

dv K(<Pv ldv\dt CD\di + ~fo)

as was found formerly, Art. 119.

SECT. VIII.] EQUATIONS APPLIED TO A SPHERE. 125

We might also suppose that particles equally distant fromthe centre have not received a common initial temperature;in this case we should arrive at a much more general equation.

156. To determine, by means of equation (A), the movementof heat in a sphere which has been immersed in a liquid, weshall regard v as a function of r and t; r is a function of x, y} z,given by the equation

r being the variable radius of an envelope. We have then

dv _dv dr , d2v _ d2v /dr\2 dv d2rdx ~~ dr dos dx2 dr2 \dx) dr dx2 f

dv dv dr , d2v d2v /dr\2 dv d2ra n d == I I -4- •

dy dr dy dy2 dr2 \dy) dr dy2'dv _dv dr , d2v __ d\ /dr\2 dv d2rdz ~~ dr dz dzl ~~ dr2 \dz) dr dz2 '

Making these substitutions in the equation

dvJ{(<Pv

we shall have

dv K Vtfv (

The equation x2 + y2 + z2 = r2 gives the following results ;

dr , - (dr\2

dr A - (dr\2 , d2ry = r-T- and 1 = I -7- ) + r - ^ ,u dy \dyj dy2

dr , - fdr\2 , d2rzz^r-j- and 1 = 1-=-) +r-rz.dz \dzj dz'

The three equations of the first order give:

fdr V fdr V , fdr\\ (d2r ,d*r d'r) .


The three equations of the second order give:

3

and substituting for

its value 1, we have

dV dV dVdx* dy2 dz2

Making these substitutions in the equation (a) we have theequation

~dt~CD W + r dry

which is the same as that of Art. 114.The equation would contain a greater number of terms, if we

supposed molecules equally distant from the centre not to havereceived the same initial temperature.

We might also deduce from the definite equation (B), theequations which express the state of the surface in particularcases, in which we suppose solida of given form to communicatetheir heat to the atmospheric air; but in most cases these equa-tions present themselves at once, and their form is very simple,when the co-ordinates are suitably chosen.

SECTION IX.

General Remarks.

157. The investigation of the laws of movement of heat insolids now consists in the integration of the equations which wehave constructed; this is the object of the following chapters.We conclude this chapter with general remarks on the natureof the quantities which enter into our analysis.

In order to measure these quantities and express them nume-rically, they must be compared with different kinds of units, five

SECT. IX.] GENERAL REMARKS. 127

in number, namely, the unit of length, the unit of time, that oftemperature, that of weight, and finally the unit which serves tomeasure quantities of heat. For the last unit, we might havechosen the quantity of heat which raises a given volume of acertain substance from the temperature 0 to the temperature 1.The choice of this unit would have been preferable in manyrespects to that of the quantity of heat required to convert a massof ice of a given weight, into an equal mass of water at 0, withoutraising its temperature. "We have adopted the last unit onlybecause it had been in a manner fixed beforehand in several workson physics; besides, this supposition would introduce no changeinto the results of analysis.

158. The specific elements which in every body determinethe measurable effects of heat are three in number, namely, theconducibility proper to the body, the conducibility relative to theatmospheric air, and the capacity for heat. The numbers whichexpress these quantities are, like the specific gravity, so manynatural characters proper to different substances.

We have already remarked, Art. 36, that the conducibility ofthe surface would be measured in a more exact manner, if we hadsufficient observations on the effects of radiant heat in spacesdeprived of air.

It may be seen, as has been mentioned in the first section ofChapter I , Art. 11, that only three specific coefficients, K} h> C,enter into the investigation; they must be determined by obser-vation; and we shall point out in the sequel the experimentsadapted to make them known with precision.

159. The number C which enters into the analysis, is alwaysmultiplied by the density D, that is to say, by the number ofunits of weight which are equivalent to the weight of unit ofvolume; thus the product CD may be replaced by the coeffi-cient c. In this case we must understand by the specific capacityfor heat, the quantity required to raise from temperature 0 totemperature 1 unit of volume of a given substance, and not unit ofweight of that substance.

With the view of not departing from the common definition,we have referred the capacity for heat to the weight and not to


the volume; but it would be preferable to employ the coefficient cwhich we have just defined; magnitudes measured by the unitof weight would not then enter into the analytical expressions:we should have to consider only, 1st, the linear dimension w, thetemperature v} and the time t; 2nd, the coefficients c, h, and K.The three first quantities are undetermined, and the three othersare, for each substance, constant elements which experimentdetermines. As to the unit of surface and the unit of volume,they are not absolute, but depend on the unit of length.

160. It must now be remarked that every undeterminedmagnitude or constant has one dimension proper to itself, andthat the terms of one and the same equation could not be com-pared, if they had not the same exponent of dimension. We haveintroduced this consideration into the theory of heat, in order tomake our definitions more exact, and to serve to verify theanalysis; it is derived from primary notions on quantities; forwhich reason, in geometry and mechanics, it is the equivalentof the fundamental lemmas which the Greeks have left us with-out proof.

161. In the analytical theory of heat, every equation (E)expresses a necessary relation between the existing magnitudesx, t, v, c, hi K. This relation depends in no respect on the choiceof the unit of length, which from its very nature is contingent,that is to say, if we took a different unit to measure the lineardimensions, the equation (E) would still be the same. Supposethen the unit of length to be changed, and its second value to beequal to the first divided by m. Any quantity whatever x whichin the equation (E) represents a certain line ab, and which, con-sequently, denotes a certain number of times the unit of length,becomes mx, corresponding to the same length ab; the value tof the time, and the value v of the temperature will not bechanged; the same is not the case with the specific elements

h, K, c: the first, h, becomes —2; for it expresses the quantity of

heat which escapes, during the unit of time, from the unit of sur-face at the temperature 1. If we examine attentively the natureof the coefficient Ky as we have defined it in Articles 68 and 135,

SECT. IX.J UNITS AND DIMENSIONS. 129

rr

we perceive that it becomes —: for the flow of heat variesm

directly as the area of the surface, and inversely as the distancebetween two infinite planes (Art. 72). As to the coefficient cwhich represents the product CD, it also depends on the unit oflength and becomes —3; hence equation (E) must undergo no

til

change when we write mx instead of x, and at the same time

— > ~~2 ? —*> instead of K, k, c ; the number m disappears afterthese substitutions : thus the dimension of x with respect to theunit of length is 1, that of K is — 1, that of h is — 2, and that of cis — 3. If we attribute to each quantity its own exponent of di-mension, the equation will be homogeneous, since every term willhave the same total exponent. Numbers such as S, which repre-sent surfaces or solids, are of two dimensions in the first case,and of three dimensions in the second. Angles, sines, and othertrigonometrical functions, logarithms or exponents of powers, are,according to the principles of analysis, absolute numbers which donot change with the unit of length ; their dimensions must there-fore be taken equal to 0, which is the dimension of all abstractnumbers.

If the unit of time, which was at first 1, becomes - , the numbern

t will become nt, and the numbers x and v will not change. Thecoefficients K. h, c will become — , •-, c. Thus the dimensions

n nof x} t, v with respect to the unit of time are 0, 1, 0, and those ofK, hy c are — 1, — 1, 0.

If the unit of temperature be changed, so that the temperature1 becomes that which corresponds to an effect other than theboiling of water; and if that effect requires a less temperature,which is to that of boiling water in the ratio of 1 to the number p;v will become vp, x and t will keep their values, and the coeffi-cients K, h, c will become —, - , - .

p p pThe following table indicates the dimensions of the three

undetermined quantities and the three constants, with respectto each kind of unit.

F. H. i)

130 THEORY OF HEAT. [CH. II. SECT. IX.

Quantity or Constant.

Exponent of dimension of x ...

„ ,, t ...

v ...

The specific conducibility, K ...

The surface conducibility, h ...

The capacity for heat, c ...

Length.

1

0

0

- 1

- 2

- 3

Duration.

0

1

0

- 1

- 1

0

Temperature.

0

0

1

- 1

- 1

1

162. If we retained the coefficients C and D, whose producthas been represented by c, we should have to consider the unit ofweight, and we should find that the exponent of dimension, withrespect to the unit of length, is — 3 for the density D, and 0for G.

On applying the preceding rule to the different equations andtheir transformations, it will be found that they are homogeneouswith respect to each kind of unit, and that the dimension of everyangular or exponential quantity is nothing. If this were not thecase, some error must have been committed in the analysis, orabridged expressions must have been introduced.

If, for example, we take equation (b) of Art. 105,

x* CDSVj

we find that, with respect to the unit of length, the dimension ofeach of the three terms is 0; it is 1 for the unit of temperature,and — 1 for the unit of time.

In the equation v = Ae~x KI of Art. 76, the linear dimen-sion of each term is 0, and it is evident that the dimension of the

exponent xA/y== is always nothing, whatever be the units of

length, time, or temperature.

CHAPTER III.

PROPAGATION OF HEAT IN AN INFINITE RECTANGULAR SOLID.

SECTION I.

Statement of the 'problem.

163. PROBLEMS relative to the uniform propagation, or tothe varied movement of heat in the interior of solids, are reduced,by the foregoing methods, to problems of pure analysis, andthe progress of this part of physics will depend in consequenceupon the advance which may be made in the art of analysis.The differential equations which we have proved contain thechief results of the theory; they express, in the most generaland most concise manner, the necessary relations of numericalanalysis to a very extensive class of phenomena; and theyconnect for ever with mathematical science one of the mostimportant branches of natural philosophy.

It remains now to discover the proper treatment of theseequations in order to derive their complete solutions and aneasy application of them. The following problem offers thefirst example of analysis which leads to such solutions; itappeared to us better adapted than any other to indicate theelements of the method which we have followed.

164. Suppose a homogeneous solid mass to be containedbetween two planes B and G vertical, parallel, and infinite, andto be divided into two parts by a plane A perpendicular to theother two (fig. 7); we proceed to consider the temperatures ofthe mass BAG bounded by the three infinite planes A, B, G.The other part B'AGf of the infinite solid is supposed to be aconstant source of heat, that is to say, all its points are main-tained at the temperature 1, which cannot alter. The two

9—2

132 THEORY OF HEAT. [CHAP. III.

lateral solids bounded, one by the plane C and the plane Aproduced, the other by the plane B and the plane A pro-

Fig. 7.

r\Cf

duced, have at all points the constant temperature 0, someexternal cause maintaining them always at that temperature;lastly, the molecules of the solid bounded by A, B and G havethe initial temperature 0. Heat will pass continually from thesource A into the solid BAG, and will be propagated there inthe longitudinal direction, which is infinite, and at the sametime will turn towards the cool masses B and 0, which will ab-sorb great part of it. The temperatures of the solid BAG willbe raised gradually : but will not be able to surpass nor evento attain a maximum of temperature, which is different fordifferent points of the mass. It is required to determine thefinal and constant state to which the variable state continuallyapproaches.

If this final state were known, and were then formed, it wouldsubsist of itself, and this is the property which distinguishesit from all other states. Thus the actual problem consists indetermining the permanent temperatures of an infinite rect-angular solid, bounded by two masses of ice B and G, and amass of boiling water A \ the consideration of such simple andprimary problems is one of the surest modes of discovering thelaws of natural phenomena, and we see, by the history of thesciences, that every theory has been formed in this manner.

1G5. To express more briefly the same problem, supposea rectangular plate BA G, of infinite length, to be heated at itsbase A, and to preserve at all points of the base a constant

SECT. I.] INFINITE RECTANGULAR SOLID. 133

temperature 1, whilst each of the two infinite sides B and G,perpendicular to the base A, is submitted also at every pointto a constant temperature 0; it is required to determine whatmust be the stationary temperature at any point of the plate.

It is supposed that there is no loss of heat at the surfaceof the plate, or, which is the same thing, we consider a solidformed by superposing an infinite number of plates similar tothe preceding : the straight line Ax which divides the plateinto two equal parts is taken as the axis of x, and the co-ordinatesof any point m are x and y; lastly, the width A of the plateis represented by 21, or, to abridge the calculation, by IT, thevalue of the ratio of the diameter to the circumference of acircle.

Imagine a point m of the solid plate BA 0, whose co-ordinatesare x and y, to have the actual temperature v, and that thequantities v, which correspond to different points, are such thatno change can happen in the temperatures, provided that thetemperature of every point of the base A is always 1, and thatthe sides B and C retain at all their points the temperature 0.

If at each point m a vertical co-ordinate be raised, equal tothe temperature v, a curved surface would be formed whichwould extend above the plate and be prolonged to infinity.We shall endeavour to find the nature of this surface, whichpasses through a line drawn above the axis of y at a distanceequal to unity, and which cuts the horizontal plane of xy alongtwo infinite straight lines parallel to x.

166. To apply the general equation

dvdt

we must consider that, in the case in question, abstraction isd2v

made of the co-ordinate z, so that the term -^ must be omitted;

with respect to the first member -j-, it vanishes, since we wish todt

determine the stationary temperatures; thus the equation which

belongs to the actual problem, and determines the propertiesof the required curved surface, is the following :

d2v d2v A , N

The function of x and y, $ (x, y), which represents the per-manent state of the solid BA G, must, 1st, satisfy the equation(a); 2nd, become nothing when we substitute - ^ ir or -f \ir for y,whatever the value of x may be ; 3rd, must be equal, to unitywhen we suppose x = 0 and y to have any value included between— \IT and + -|7r.

Further, this function </> (x, y) ought to become extremelysmall when we give to x a very large value, since all the heatproceeds from the source A.

167. In order to consider the problem in its elements, weshall in the first place seek for the simplest functions of xand y, which satisfy equation (a); we shall then generalise thevalue of v in order to satisfy all the stated conditions. By thismethod the solution will receive all possible extension, and weshall prove that the problem proposed admits of no othersolution.

Functions of two variables often reduce to less complex ex-pressions, when we attribute to one of the variables or to bothof them infinite values; this is what may be remarked in alge-braic functions which, in this particular case, take the form ofthe product of a function of # by a function of y.

We shall examine first if the value of v can be representedby such a product; for the function v must represent the stateof the plate throughout its whole extent, and consequently thatof the points whose co-ordinate x is infinite. We shall thenwrite v = F{x)f(y); substituting in equation (a) and denoting

^ & by F" (x) and ^ b y / " (y), we shall have

we then suppose -y^j = m and *-~^j ~-m, m being any

SECT. I.] INFINITE RECTANGULAR PLATE. 135

constant quantity, and as it is proposed only to find a particularvalue of v, we deduce from the preceding equations F(x) = e~mx,f(y) = cos my.

168. We could not suppose m to be a negative number,and we must necessarily exclude all particular values of v, intowhich terms such as emx might enter, m being a positive number,since the temperature v cannot become infinite when x is in-finitely great. In fact, no heat being supplied except from theconstant source A, only an extremely small portion can arriveat those parts of space which are very far removed from thesource. The remainder is diverted more and more towards theinfinite edges B and G, and is lost in the cold masses whichbound them.

The exponent m which enters into the function e~mx cos mijis unknown, and we may choose for this exponent any positivenumber: but, in order that v may become nul on makingy = — \ 7r or y = + f 7r, whatever x may be, m must be takento be one of the terms of the series, 1, 3, 5, 7, &c.; by thismeans the second condition will be fulfilled.

169. A more general value of v is easily formed by addingtogether several terms similar to the preceding, and we have

v = ae~x cos y + be'3* cos 3y + ce~5xcos 5y + de~7x cos 7y + &c (b).

It is evident that the function v denoted by <f> (x, y) satisfiesd2v d2v

the equation -r—2 + -^ = 0, and the condition (f>(x, ± \ if) = 0.

A third condition remains to be fulfilled, which is expressed thus,$ (0, y) = 1, and it is essential to remark that this result mustexist when we give to y any value whatever included between— \ ir and + \ IT. Nothing can be inferred as to the valueswhich the function </> (0, y) would take, if we substituted in placeof y a quantity not included between the limits — \ ir and + \ IT.Equation (b) must therefore be subject to the following condition :

1 = a cos y + b cos Sy + c cos hy + d cos 7y + &c.The coefficients, a, 6, c, d, &c, whose number is infinite, are

determined by means of this equation.The second member is a function of y, which is equal to 1


so long as the variable y is included between the limits - f 7rand +J7r. It may be doubted whether such a function exists,but this difficulty will be fully cleared up by the sequel.

170. Before giving the calculation of the coefficients, wemay notice the effect represented by each one of the terms ofthe series in equation (b).

Suppose the fixed temperature of the base A, instead ofbeing equal to unity at every point, to diminish as the pointof the line A becomes more remote from the middle point,being proportional to the cosine of that distance; in this caseit will easily be seen what is the nature of the curved surface,whose vertical ordinate expresses the temperature v or cf> (x, y).If this surface be cut at the origin by a plane perpendicularto the axis of x, the curve which bounds the section will havefor its equation v = a cos y; the values of the coefficients willbe the following :

a = a, 5=0 , c = 0 , ^ = 0 ,

and so on, and the equation of the curved surface will be

v — ae~x cos y.

If this surface be cut at right angles to the axis of y, thesection will be a logarithmic spiral whose convexity is turnedtowards the axis; if it be cut at right angles to the axis of xy

the section will be a trigonometric curve whose concavity isturned towards the axis.

It follows from this that the function -=-^ is always positive,ctx

d2vand - -2 i s always negative. Now the quantity of heat which

a molecule acquires in consequence of its position between two

others in the direction of x is proportional to the value of —dx*

(Art. 123) : it follows then that the intermediate molecule receivesfrom that which precedes it, in the direction of x, more heat thanit communicates to that which follows it. But, if the same mole-cule be considered as situated between two others in the directionof y, the function j ~ 2 being negative, it appears that the in-

SECT.IL] TRIGONOMETRIC SERIES. 337

termediate molecule communicates to that which follows it moreheat than it receives from that which precedes it. Thus itfollows that the excess of the heat which it acquires in the direc-tion of xy is exactly compensated by that which it loses in the

direction of y, as the equation -,~2 + _? = 0 denotes. Thus

then the route followed by the heat which escapes from thesource A becomes known. It is propagated in the directionof x, and at the same time it is decomposed into two parts,one of which is directed towards one of the edges, whilst theother part continues to separate from the origin, to be decomposedlike the preceding, and so on to infinity. The surface whichwe are considering is generated by the trigonometric curve whichcorresponds to the base Ay moved with its plane at right angles tothe axis of x along that axis, each one of its ordinates de-creasing indefinitely in proportion to successive powers of thesame fraction.

Analogous inferences might be drawn, if the fixed tempera-tures of the base A were expressed by the term

b cos Sy or c cos 5y, &c.;

and in this manner an exact idea might be formed of the move-ment of heat in the most general case; for it will be seen bythe sequel that the movement is always compounded of a multi-tude of elementary movements, each of which is accomplishedas if it alone existed.

SECTION II.

First example of the use of trigonometric series in the theoryof heat*

171. Take now the equation

1 = a cos y + b cos Sy + c cos 5y + d cos 7y + &c,

in which the coefficients a, b, c, d, &c. are to be determined.In order that this equation may exist, the constants must neces-

138 THEOEY OF HEAT. [CHAP. III.

sarily satisfy the equations which are obtained by successivedifferentiations; whence the following results,

1 = a cos y + b cos Sy 4- c cos 5y + d cos 7y + &c,

0 = a sin y + 35 sin 3y + 5c sin 5y + 7d sin 7y + &c,

0 = a cos 2/ + 326 cos 3z/ + 52c cos by + Td cos 7y + &c,

0 = a sin y + 336 sin Sy + 53c sin hy + 73i sin 7y + &c,

and so on to infinity.

These equations necessarily hold when y = 0, thus we have

1 = a + 5+ c+ d + e + / + # + . . . & c ,

0 = a + 326 + 52c + 72i + 92^ + l l a / + ... Ac,0 = a + S'b + 54c + 7\l+9*e + ... &c,0 = a-f 365 + 56c + 766?+ ... &c,0 = a + 386 + 58c + ... &c,&c.

The number of these equations is infinite like that of theunknowns a, b, c, d, e, ... &c. The problem consists in eliminatingall the unknowns, except one only.

172. In order to form a distinct idea of the result of theseeliminations, the number of the unknowns a, &, c, d, ... &c, willbe supposed at first definite and equal to m. We shall employthe first m equations only, suppressing all the terms containingthe unknowns which follow the m first. If in succession mbe, made equal to 2, 3, 4, 5, and so on, the values of the un-knowns will be found on each one of these hypotheses. Thequantity a, for example, will receive one value for the caseof two unknowns, others for the cases of three, four, or successivelya greater number of unknowns. It will be the same with theunknown b, which will receive as many different values as therehave been cases of elimination; each one of the other unknownsis in like manner susceptible of an infinity of different values.Now the value of one of the unknowns, for the case in whichtheir number is infinite, is the limit towards which the valueswhich it receives by means of the successive eliminations tend.It is required then to examine whether, according as the numberof unknowns increases, the value of each one of a, 6, c, d ... &c.does not converge to a finite limit which it continually ap-proaches.

SECT. II.J DETERMINATION OF COEFFICIENTS. 139

Suppose the six following equations to be employed:

1 = a + b + c + d + e + f + &c,0 = a + 32& + 52c + Td + 92e + l l 2 / + &c,

0 = a + 36Z> + 56c + 7V + 96e + l l 6 / + &c,

0 = a + S8b + 58c + 78d + 98^ + l l 8 / + &c ,

0 = a + 3106 + 510c + Td + Q10e + l l 1 0 / + &c.

The five equations which do not contain/are :

i r = a ( l l 2 - l 2 ) + J(H2-32)+ c(H2-52)+

0 =a ( l l 2

0=a(ll2-r)+366(ll2-32)+56c(ir-52)+76d(ll2-72)+9^(ll2-92) ,0=a(ll2-l2)+386(ir-32)+58c(ll2-52)+78d(ir--72)+98e(ll2--92).

Continuing the elimination we shall obtain the final equationin a, which is :

a (ll2 -12) (92 -12) (72 -12) (52 -12) (32 -12) = l l 2 . 9 2 . 72. 52. 32.12.

173. If we had employed a number of equations greaterby unity, we should have found, to determine a, an equationanalogous to the preceding, having in the first member onefactor more, namely, 132—I2, and in the second member 132

for the new factor. The law to which these different values ofa are subject is evident, and it follows that the value of a whichcorresponds to an infinite number of equations is expressed thus:

3 2 5 2 72 _9^ l l 2

a ~ 32 - 1 2 ' s2 - r * 72 - r2 * s2 - r * i r - r *_ 3 . 3 5 . 5 7 . 7 9 . 9 1 1 . 1 1 „

O r a " 2 . 4 * 4 . 6 ' 6 . 8 * 8 . 1 0 1 0 . 1 2 ^ a

Now the last expression is known and, in accordance with4

Wallis' Theorem, we conclude that a = —. I t is required then

only to ascertain the values of the other unknowns.

174. The five equations which remain after the eliminationof / may be compared with the five simpler equations whichwould have been employed if there had been only five unknowns.

140 THEOKY OF-HEAT. [CHAP. III.

The last equations differ from the equations of Art. 172, inthat in them e, d, c, b, a are found to be multiplied respec-tively by the factors

I I 2 - 9 2 1 P - 7 2 l l 2 - 5 2 II2 - S 2 I I 2 - ! 2

i i 5 " 1 i i 2 ' i i 2 ' i r ' i r '

It follows from this that if we had solved the five linearequations which must have been employed in the case of fiveunknowns, and had calculated the value of each unknown, itwould have been easy to derive from them the value of theunknowns of the same name corresponding to the case in whichsix equations should have been employed. I t would suffice tomultiply the values of e, d, c, b, ay found in the first case, by theknown factors. It will be easy in general to pass from the valueof one of these quantities, taken on the supposition of a certainnumber of equations and unknowns, to the value of the samequantity, taken in the case in which there should have beenone unknown and one equation more. For,example, if the valueof ey found on the hypothesis of five equations and five unknowns,is represented by E, that of the same quantity, taken in the case

II2

of one unknown more, will be E--——2. The same value,j . J. — fj

taken in the case of seven unknowns, will be, for the same reason,

F II2 132

l l 2 - 9 2 1 3 2 - 9 2 '

and in the case of eight unknowns it will be

II2 1 3 2 15 2

Eir-92'13*-92:i52-92'

and so on. In the same manner it will suffice to know thevalue of b, corresponding to the case of two unknowns, to derivefrom it that of the same letter which corresponds to the casesof three, four, five unknowns, &c. We shall only have to multiplythis first value of b by

5 2 - 3 2 ' 7 2 - 3 2 * W^92

SECT. II.J DETERMINATION OF COEFFICIENTS. 141

Similar ly if we knew t h e value of c for t he case of threeunknowns , we should mul t ip ly this value by the successive factors

72 92 I I 2

W e should calculate t h e value of d for t h e case of four unknownsonly, and mult iply th is value by

92 II2 132

The calculation of the value of a is subject to the same rule,for if its value be taken for the case of one unknown, and multi-plied successively by

T 9*3 a _ l " 5 2 - l 2 ' 7 2 - l 2 ' 9 2 - l 2 '

the final value of this quantity will be found.

175. The problem is therefore reduced to determining thevalue of a in the case of one unknown, the value of b in the caseof two unknowns, that of c in the case of three unknowns, and soon for the other unknowns.

It is easy to conclude, by inspection only of the equations andwithout any calculation, that the results of these successive elimi-nations must be

a = 1,

I2

b =

C = :

l 2 - 3 2 '

I2 3 2

I2 2*d = _ 72 • 32 _ 72 • g 2 _ 72 ,

I 2 32 52 /r-92 ' 3 2 -9 2 ' 5 z - i f ' 72-92*

176. It remains only to multiply the preceding quantities bythe series of products which ought to complete them, and whichwe have given (Art. 174). We shall have consequently, for the

final values of the unknowns a, b, c, d, e, f, &c, the followingexpressions :

32 52 T 98 II2 ,** -*• • Q2 1 2 • r 2 12 • 72 12 * Q'2 1 2 ' 1 1 2 "l 2

_ l 2 5 ' 7 2 9 2 1 1 * „6 - p ^ • 5^_ 32 • 7* _ 3 2 • 9 2 - 3 2 ' H 2 - 3 2 '

c = -r,I2 3 2 72 92 I I 2

-52 • 33_52 " 72-52 ' 92-52 ' i r - 5 :;&C,

I2 32 52 92 I I 2 g

1 — 7 o — 1 o — t y — { 1 1 — 7

l ! _ 32 52 72 I I 2 132 .e ~ l2-92 ' 32-92 ' 52-92 ' 72-92 ' 1F-92 ' 132-92 '

I 2 3 2 52 72 92 13 2

J- r _ H 2 • 3 2 - l l 2 • 5 2 - l l 2 ' 7*-II" ' 9 2 - l l 2 ' 13 2 -11 2

or,

, . 3 . 3 5 .5 7 . 7 ^0 + 1 & C

2 . 4 " 2 . 8 " 4 . 10 ' 6 .12 '

1 . 1 3 . 3 7 . 7 9 . 9 1 1 . 1 1 .s* z=z -X- A T P

+ 4 . 6 < 2 . 8 ' 2 . 1 2 " 4 . 1 4 " 6.16 '

<j = - h l 3 - 3 5 .5 9^9 11.116 . 8 ' 4 .10 ' 2 .12 " 2716 ' "4Tl8" '

, 1 1 3 ^ l^p_ J7V7_ 11.11 13.13 g

8 .10 ' 6 .12 ' 4 . 14 " 2 . 16 ' 2 . 20 * ^722~ '

f = 1-1 3 . 3 5 .5 7 . 7 9 . 9 13 .13 1 5 . 1 5 .•' 10.12 # 8 .14 ' 6 .16 " 4 .18 ' 2 . 20 " 2 . 24 " 1726

The quantity \TT or a quarter of the circumference is equiva-lent, according to Wallis' Theorem, to

2 . 2 4 . 4 (L6 8jj3 10.10 12.12 14 141.3 • 3 .5 - 5 . 7 - 7 . 9 T T l T • 1X713 * 1 3 7 l 5 & a

SECT. II.] VALUES OF THE COEFFICIENTS. 143

If now in the values of a, b, c, d, &c, we notice what are thefactors which must be joined on to numerators and denominatorsto complete the double series of odd and even numbers, we findthat the factors to be supplied are:

9

3 . 3for b

for c

6 '5 .5

, , 7.7TOr d y r - ,

. 9.9for e - - ,

for /11.11

22 ' j

• whence we conclude ,

ct = A •7T

J = ~ 2 - 3 ^ '

e = 2 .

9TT'

/--2.

177. Thus the eliminations have been completely effected,and the coefficients a, b, c, d, &c, determined in the equation

1 = a cos y -\-b cos 32/ + c cos 5y -\- d cos 7y •

The substitution of these coefficients gives the following equa-tion :

- = cos y — n cos 3y -f ^ cos 5y — ^ cos 72/ + ^ cos — &c.2

The second member is a function of y, which does not changein value when we give to the variable y a value included between— \ir and -f i7r. It would be easy to prove that this series isalways convergent, that is to say that writing instead of y anynumber whatever, and following the calculation of the coefficients,we approach more and more to a fixed value, so that the differenceof this value from the sum of the calculated terms becomes lessthan any assignable magnitude. Without stopping for a proof,

1 It is a little better to deduce the value of b in a, of c in &, &c. [K. L. E.]2 The coefficients a, b, c, &c, might be determined, according to the methods

of Section vi., by multiplying both sides of the first equation by cos y, cos By,

cos5v, &c, respectively, and integrating from - -IT to -\-^TT9 as was done by

D. F. Gregory, Cambridge, Mathematical Journal\ Vol. i. p. 106. [A. F.]


which the reader may supply, we remark that the fixed valuewhich is continually approached is îr, if the value attributedto y is included between 0 and îrt but that it is — ^TT, if y isincluded between \ir and § IT ; for, in this second interval, eachterm of the series changes in sign. In general the limit of theseries is alternately positive and negative ; in other respects, theconvergence is not sufficiently rapid to produce an easy approxima-tion, but it suffices for the truth of the equation.

178. The equation

y == cos x — = cos Sx + •= cos Zx — = cos 7# + &c,O O 4

belongs to a line which, having x for abscissa and y for ordinate, iscomposed of separated straight lines, each of which is parallel tothe axis, and equal to the circumference. These parallels aresituated alternately above and below the axis, at the distance îr,and joined by perpendiculars which themselves make part of theline. To form an exact idea of the nature of this line, it must besupposed that the number of terms of the function

cos x — ~ cos Sx + - cos 5x — &c.o 5

has first a definite value. In the latter case the equation

y = cos x — - cos 3x + - cos 5x — &c.o o

belongs to a curved line which passes alternately above and belowthe axis, cutting it every time that the abscissa x becomes equalto one of the quantities

1 3 5

According as the number of terms of the equation increases, thecurve in question tends more and more to coincidence with thepreceding line, composed of parallel straight lines and of perpen-dicular lines; so that this line is the limit of the different curveswhich would be obtained by increasing successively the number ofterms.

SECT. III.] ItEMAKKS ON THE SERIES. 145

SECTION III.

s on these series.

179. We may look at the same equations from another pointof view, and prove directly the equation

7T 1 _ 1 K 1 _ 1 ^ O-r = COS X — - COS 3x -f - COS OX — COS (X + - COS VX — &C.4 6 o 7 y

The case where # is nothing is verified by Leibnitz' series,

IT . 1 1 1 1 -1 + + &

We shall next assume that the number of terms of the series

cos x — - cos 5x + - cos 5x — = cos 7x + &c.S o l

instead of being infinite is finite and equal to m. We shall con-sider the value of the finite series to be a function of x and m.We shall express this function by a series arranged according tonegative powers of m\ and it will be found that the value ofthe function approaches more nearly to being constant and inde-pendent of x, as the number m becomes greater.

Let y be the function required, which is given by the equation

y = cos x — Q cos 3x + - cos ox — = cos 7x+... — cos (2m — 1) x}O D i Alii — 1

m, the number of terms, being supposed even. This equationdifferentiated with respect to x gives

dii . o . w . *-— ~ = sin x — sm Sx + sm o^ — sin 7x+ ...

ax

+ sin (2m — 3) x — sin (2?^ — 1) x ;

multiplying by 2 sin 2.r, we have

— 2 -7- sin 2^ = 2 sin x sin 2JC — 2 sin 3x sin 2o? + 2 sin 5a; sin 2^ ...a#

+ 2 sin (2m — 3) a? sin 2 ^ - 2 sin (2m — 1) a? sin 2^.

F. H. 10

14)6 THEORY OF HEAT. [CHAP. III.

Each term of the second member being replaced by thedifference of two cosines, we conclude that

- 2 -£ sin 2x = cos (- x) - cos Sxax

— cos x + cos ox

4- cos 3a; - cos 7x

— cos 5% + cos 9x

+ cos (2m — 5) x - cos (2m —l)x

- cos (2m - 3a?) + cos (2m + 1) #.

The second member reduces to

cos (2m + 1) x - cos (2m - 1 ) x} or - 2 sin 2mx sin x;

. 1 /7 , sin 2mx\hence y = 5 \[dx 1

J 2]\ coax J

1/7= 5 \[d

2]\

180. We shall integrate the second member by parts, dis*tinguishing in the integral between the factor sin 2mx dx which

must be integrated successively, and the factor or sec x° J cos x

which must be differentiated successively; denoting the resultsof these differentiations by sec' x} sec" or, sec'" x,... &c, we shallhave

2y — const. — — cos 2mx sec x + 7 —r, sin 2mx sec' x

— j—g cos 2mx sec" x H- &c.;

thus the value of y or

cos x - ;5 cos 3x + - cos 5x — s cos 7x + ... — cos (2m -1 ) .r,

which is a function of # and m, becomes expressed by an infiniteseries; and it is evident that the more the number m increases,the more the value of y tends to become constant. For thisreason, when the number m is infinite, the function y has adefinite value which is always the same, whatever be the positive

SECT. III.] PARTICULAR CASES. 147

value of xy less than |7r. NOW, if the arc x be supposed nothing,we have

which is equal to \ir. Hence generally we shall have

7 7T = COS X — j : COS Sx + ~z COS OX — = COS 7 # + 7: COS $X — &C. . . . (b).* 6 o J 7 y

181. If in this equation we assume ^ = ^ 5-, we find

- ? L - - i , 1 1 1 1 l j 1 . 02J2" 3 5""7 + 9 + n " l 3 i5 + - * c - >

by giving to the arc a? other particular values, we should findother series, which it is useless to set down, several of whichhave been already published in the works of Euler. If wemultiply equation (b) by dx, and integrate it, we have

7TX . 1 . o 1 . r 1 . ,_ 0

—- = sin x — ^ sin ox + ^ sin ox — -=:> sin 7x + &e.

Making in the last equation x = |r7r, we find7T.2 1 1 1 1

a series already known. Particular cases might be enumeratedto infinity; but it agrees better with the object of this workto determine, by following the same process, the values of thedifferent series formed of the sines or cosines of multiple arcs.

182. Let

y — sin x — ^ sin 2x + - sin 3.?; — j sin 4^ ...It O Tb

+ —'•—7 sin (m — 1) x sin mx%

m — 1 v ' m

vi being any even number. We derive from this equation

f- = cos x — cos 2-r + cos ?>x — cos 4tx •.. + cos (m — 1) x — cos 7?^;

10—2


multiplying by 2 sin av and replacing each term of the secondmember by the difference of two sines, we shall have

2 sin x -T- = sin (x + x) - sin (x - #C&JC

- sin (2a? + a?) + sin (2x - x)

+ sin (Sx + x)- sin (Sx - x)

4- sin {(m - 1) x - x) - sin {(»i + 1) a - a?}

— sin (m# + .r) + sin (mx — a?);

and, on reduction,

dy . . / >2 sm a? -~ = sin a? + sin mx — sin {nix + x):

the quantity sin mx — sin (mx 4- x)f

or sin (ma? + -|a? — \x) — sin (ma? -f \x + ^.T),

is equal to — 2 sin ^# cos (??ia? + J ^ ) ;

we have therefore

dy 1 sin A x , 1 N-y = - — . cos [mx + hx),dx 2i sin x

d\f _ 1 cos (mx 4- •£#)° r fl& ~ 2 2cos i / ; ;

whence we conclude

y=lx-

If we integrate this by parts, distinguishing between the

factor —-j- or sec J.r, which must be successively differentiated,COS o X

and the factor cos (ma? 4-ia?), which is to be integrated severaltimes in succession, we shall form a series in which the powers

of m + g enter into the denominators. As to the constant it

is nothing, since the value of y begins with that of x.

SECT. Hi.] SPECIAL SEKIES. 149

I t follows from this that the value of the finite series

sin x — 5 sin 2x + „ sin Sx — -=• sin 5x + = sin 7x — ... sin mx

differs very little from that of \x, when the number of termsis very great; and if this number is infinite, we have th,e knownequation

^ x = sin x — jr sin 2x + ^ sin Sx — -? sin 4*x + -? sin 5x — &c.

From the last series, that which has been given above forthe value of \TT might also be derived.

183, Let now

y = g> cos 2cc — j cos 4E + - cos 6x — ...^ 4* o

C0S

Differentiating, multiplying by 2 sin 2x, substituting thedifferences of cosines, and reducing, we shall have

COS X

« f i ± f 7 sin (2m + 1) xor Zy — c— lax tan x + lax —— — ;J J J cos a? '

integrating by parts the last term of the second member, and

supposing m infinite, we have y = c + ^ log cos or. If in the

equation1 1 1 1

y = jr cos 2x — 7 cos 4<x + - cos 6x — - cos 8OJ + ... &c,i» 4 0 o

we suppose x nothing, we find

1 1 1 1 1

1 i

therefore y = - log 2 + ^ log cos #\

Thus we meet with the series given by Euler,

log (2 cos \x) = cos x - ^ cos 2# f ^ cos 3 ^ - 7 c o s ^x + &c»


184. Applying the same process to the equation

y = sin x+ - sin Sx + - sin 5x + - sin 7x + &c,J 3 o 7

we find the following series, which has not been noticed,

7T = sin x + r> sin Sx + - sin ox + - sin 7x + - sin 9x + &C.14 o o / ^

It must be observed with respect to all these series, thatthe equations which are formed by them do not hold exceptwhen the variable x is included between certain limits. Thusthe function

cos x — -s cos Sx + - cos ox - ^ cos 7x + &c.3 o /

is not equal to \iry except when the variable x is containedbetween the limits which we have assigned. It is the samewith the series

sin x — -- sin 2% + - sin 3x — T sin &x+ - sin 5# — &c.2 3 4 o

This infinite series, which is always convergent, has the value\x so long as the arc x is greater than 0 and less than 7r. Butit is not equal to \xy if the arc exceeds ir\ it has on the contraryvalues very different from \x; for it is evident that in the in-terval from x = IT to x = 2ir, the function takes with the contrarysign all the values which it had in the preceding interval fromx = 0 to x = 77\ This series has been known for a long time,but the analysis which served to discover it did not indicatewhy the result ceases to hold when the variable exceeds 7r.

The method which we are about to employ must thereforebe examined attentively, and the origin of the limitation to whicheach of the trigonometrical series is subject must be sought.

185. To arrive at it, it is sufficient to consider that thevalues expressed by infinite series are not known with exactcertainty except in the case where the limits of the sum of theterms which complete them can be assigned; it must thereforebe supposed that we employ only the first terms of these series,

1 This may bo derived by integration from 0 to TT as in Art. 222. [R. L. E.J

SECT. III.] LIMITS OF THE REMAINDER. 151

and the limits between which the remainder is included mustbe found.

We will apply this remark to the equation

1 o 1 1y — cos x — - cos Sx + - cos ox — ^ cos 7x ...

o D icos (2m — 3)x cos (2m — 1) x

+ 2m - 3 2m -1 *The number of terms is even and is represented by m; from it. , . -. ,, . 2dy sin 2m# t . r .is derived the equation —— = , whence we may infer the

(XX COS X

value of yy by integration by parts. Now the integral Juvdxmay be resolved into a series composed of as many terms asmay be desired, u and v being functions of x. We may write, forexample,

\iLvdx = c + u lvdx — -j- \dx Ivdx -h-j-^ldxl dxlvdx

an equation which is verified by differentiation.

Denoting sin 2mx by v and sec x by u, it will be found that

2y = c — — sec x cos 2?ra +92—2sec'^ ^n %mx + or-s~sec"# c o s

sec" a; o

2W" • C0S 2

186. It is required now to ascertain the limits between which

the integral --3—31 [d (sec" x) cos 2mx} which completes the series

is included. To form this integral an infinity of values mustbe given to the arc x, from 0, the limit at which the integralbegins, up to x} which is the final value of the arc; for each oneof these values of x the value of the differential d (sec" x) mustbe determined, and that of the factor cos 2mx, and all the partialproducts must be added: now the variable factor cos 2mx isnecessarily a positive or negative fraction; consequently theintegral is composed of the sum of the variable values of thedifferential c7(scc'rcr), multiplied respectively by these fractions.

152 THEOKY OF HEAT. [CHAP. IIL

The total value of the integral is then less than the sum of thedifferentials d (sec" a*), taken from x = 0 up to cr, and it is greaterthan this sum taken negatively: for in the first case we replacethe variable factor cos 2mx by the constant quantity 1, and inthe second case we replace this factor by — 1 : now the sum ofthe differentials d(sec" x), or which is the same thing, the integralfd (sec" x), taken from x = 0, is sec" x — sec" 0 ; sec" x is a certainfunction of x, and sec"O is the value of this function, taken onthe supposition that the arc x is nothing.

The integral required is therefore included between+ (sec".*; — sec" 0) and — (sec" x — sec" 0);

that is to say, representing by h an unknown fraction positive ornegative, we have always

/ [d (sec" x) cos 2mx) = Jc (sec" x - sec" 0).Thus we obtain the equation

%j = c - —- sec x cos 2mx + -—^ sec' x sin 2mx + ——3 sec"a?cos2ww;

+ ^ ^ (sec"a-sec"0),

in which the quantity -^—3 (sec" x — sec" 0) expresses exactly theZi Tib

sum of all the last terms of the infinite series.

187. If we had investigated two terms only we should havehad the equation

2y = c- — secarcos 2mx 4-^—z sec'xsin 2?nx + -2 (sec'x-sec'0).Jallls Zi 111 2i 711

From this it follows that we can develope the value of y in asmany terms as we wish, and express exactly the remainder ofthe series; we thus find the set of equations

2y = c 1

12 m

- sec x cos 2m,i

-secxcos2mxi

X±±Ai

sec x — se

sec' x sin

:CO),

k'I'm

C'x ~ sec' ° '2y = c--— secxcos2mx+ -—<,sec'xsin 2mx + -=-^ sec"x cos2mx

+ ^ 3 (sec" x - s e c " 0).

SECT. III.] LIMITS OF THE VARIABLE. 153

The number h which enters into these equations is not thesame for all, and it represents in each one a certain quantitywhich is always included between 1 and — 1; m is equal to thenumber of terms of the series

cos x — ^ cos Sx + - cos 5x — ... — = cos (2m — 1) x,3 5 Zm — 1 ^ '

whose sum is denoted by y.

188. These equations could be employed if the number mwere given, and however great that number might be, we coulddetermine as exactly as we pleased the variable part of the valueof y. If the number m be infinite, as is supposed, we considerthe first equation only; and it is evident that the two termswhich follow the constant become smaller and smaller; so thatthe exact value of 2y is in this case the constant c; this constantis determined by assuming x = 0 in the value of y, whence weconclude

- = cos x — - cos Sx + - cos 5x — = cos 7x + 7T cos 9x — &c.4 3 o 7 9

It is easy to see now that the result necessarily holds if thearc x is less than \ir. . In fact, attributing to this arc a definitevalue X as near to \ir as we please, we can always give to m

a value so great, that the term -— (sec a; — sec 0), which completes

the series, becomes less than any quantity whatever; but theexactness of this conclusion is based on the fact that the termsec x acquires no value which exceeds all possible limits, whenceit follows that the same reasoning cannot apply to the case inwhich the arc x is not less than \ir.

The same analysis could be applied to the series which expressthe values of \xy log cos x, and by this means we can assignthe limits between which the variable must be included, in orderthat the result of analysis may be free from all uncertainty;moreover, the same problems may be treated otherwise by amethod founded on other principles1.

189. The expression of the law of fixed temperatures ina solid plate supposed the knowledge of the equation

1 Cf. De Morgan's Diff. and Int. Calculus, pp. 605—6C0. [A. F.]


^ = cos x - - cos ox + - cos ox - ^ cos 7x + ^ cos 9a; - &c.4 3 o / y

A simpler method of obtaining this equation is as follows :

If the sum of two arcs is equal to \ir, a quarter of thecircumference, the product of their tangent is 1; we have there-fore in general

^TT = arc tan it + arc t a n - (c) ;

the symbol arc tan u denotes the length of the arc whose tangentis u, and the series which gives the value of that axe is wellknown ; whence we have the following result:

« •

If now we write exs!~l instead of u in equation (c), and in equa-tion (d), we shall have

s 7r = arc tan ex^~l + arc tan e~x^~1y

and - ir = cos x — - cos Sx + - cos ox — = cos 7x + - cos 9x — &c.4 6 o ( v

The series of equation (d) is always divergent, and that ofequation (b) (Art. 180) is always convergent; its value is \iror — \TT.

SECTION IV.

General solution.

190. We can now form the complete.solution of the problemwhich we have proposed; for the coefficients of equation (b)(Art. 1C9) being determined, nothing remains but to substitutethem, and we have

— = e"x cos y - - e~?lX cos oy -4- ^ e"bx cos 5y ~ ^ e"7x cos 7y + &c.... (a).

SECT. IV.] COEXISTENCE OF PARTIAL STATES. 155

This value of v satisfies the equation -=—2 + -=-, = 0; it becomesctx dy

nothing when we give to y a value equal to \ir or — \ir; lastly,it is equal to unity when x is nothing and y is included between— \ir and + hir. Thus all the physical conditions of the problemare exactly fulfilled, and it is certain that, if we give to eachpoint of the plate the temperature which equation (a) deter-mines, and if the base A be maintained at the same time at thetemperature 1, and the infinite edges B and G at the tempera-ture 0, it would be impossible for any change to occur in thesystem of temperatures.

191. The second member of equation (a) having the formof an exceedingly convergent series, it is always easy to deter-mine numerically the temperature' of a point whose co-ordinatesx and y are known. The solution gives rise to various resultswhich it is necessary to remark, since they belong also to thegeneral theory.

If the point m, whose fixed temperature is considered, is verydistant from the origin A, the value of the second member ofthe equation (a) will be very nearly equal to e~x cos y; it reducesto this term if x is infinite.

4The equation v = - e~x cos y represents also a state of the

solid which would be preserved without any change, if it wereonce formed; the same would be the case with the state repre-

4scnted by the equation v = — ezx cos 3y, and in general eachterm of the series corresponds to a particular state which enjoysthe same property. All these partial systems exist at once inthat which equation (a) represents; they are superposed, andthe movement of heat takes place with respect to each of themas if it alone existed. In the state which corresponds to anyone of these terms, the fixed temperatures of the points of thebase A differ from one point to another, and this is the only con-dition of the problem which is not fulfilled; but the general statewhich results from the sum of all the terms satisfies this specialcondition.

According as the point whoso temperature is considered is

156 THEOKY OF HEAT. [CHAP. III.

more distant from the origin, the movement of heat is less com-plex: for if the distance x is sufficiently great, each term ofthe series is very small with respect to that which precedes it,so that the state of the heated plate is sensibly represented bythe first three terms, or by the first two, or by the first only,for those parts of the plate which are more and more distantfrom the origin.

The curved surface whose vertical ordinate measures thefixed temperature v, is formed by adding the ordinates of amultitude of particular surfaces whose equations are

—-1 =excosy, -j1 = —\e cos3y, -^ = e cos5yy &c.

The first of these coincides with the general surface when xis infinite, and they have a common asymptotic sheet.

If the difference v—v1 of their ordinates is considered to bethe ordinate of a curved surface, this surface will coincide, when xis infinite, with that whose equation is \rrrv2 — — \e~Zx cos 3y. Allthe other terms of the series produce similar results.

The same results would again be found if the section at theorigin, instead of being bounded as in the actual hypothesis bya straight line parallel to the axis of yy had any figure whateverformed of two symmetrical parts. It is evident therefore thatthe particular values

ae~xcosy, be~3x cos 3y, ce"oX cos 5y, &c,

have their origin in the physical problem itself, and have anecessary relation to the phenomena of heat. Each of themexpresses a simple mode according to which heat is establishedand propagated in a rectangular plate, whose infinite sides retaina constant temperature. The general system of temperaturesis compounded always of a multitude of simple systems, and theexpression for their sum has nothing arbitrary but the coeffi-cients a, by c, d, &c.

192. Equation (a) may be employed to determine all thecircumstances of the permanent movement of heat in a rect-angular plate heated at its origin. If it be asked, for example,what is the expenditure of the source of heat, that is to say,

SECT. IV.] EXPENDITURE OF THE SOURCE OF HEAT. 157

what is the quantity which, during a given time, passes acrossthe base A and replaces that which flows into the cold massesJB and C) we must consider that the flow perpendicular to the

axis of y is expressed by —K-j-. The quantity which during

the instant dt flows across a part dy of the axis is therefore

and, as the temperatures are permanent, the amount of the flow,

during unit of time, is — K-j-dy. This expression must be

integrated between the limits y = — iir and y = + \TT, in orderto ascertain the whole quantity which passes the base, or whichis the same thing, must be integrated from y = 0 to y = \iry and

the result doubled. The quantity -y- is a function of x and y,

in which x must be made equal to 0, in order that the calculationmay refer to the base A, which coincides with the axis of y. Theexpression for the expenditure of the source of heat is there-fore 2 I (— K j - dy J, The integral must be taken from y — 0 to

dvy = ITT ; if, in the function -y- , x is not supposed equal to 0,

but x = x, the integral will be a function of x which will denotethe quantity of heat which flows in unit of time across a trans-verse edge at a distance x from the origin.

193. . If we wish to ascertain the quantity of heat which,during unit of time, passes across a line drawn on the plate

parallel to the edges B and C, we employ the expression — ^~T~ >

and, multiplying it by the element dx of the line drawn, integratewith respect to x between the given boundaries of the line; thus

the integral If — K ~r dx\ shews how much heat flows across the6 J l dy J

whole length of the line ; and if before or after the integrationwe make 2/ = |7r, we determine the quantity of heat which, duringunit of time, escapes from the plate across the infinite edge C.We may next compare the latter quantity with the expenditure


of the source of heat; for the source must necessarily supplycontinually the heat which flows into the masses B and (7. Ifthis compensation did not exist at each instant, the system oftemperatures would be variable,

194. Equation (a) gives

- K -J- = — (e"* cosy - e~3x cos 3y + e~5x cos 5y - e'lx cos 7y + &c);

multiplying by dy, and integrating from y = 0, we have

—- (e~x siny — - e~Zx sin 3y + - e"cX sin oy — = e~lx sin 7# + &c. J .

If y be made =-|7r, and the integral doubled, we obtain

E(e + le + ^ + ^ + &

as the expression for the quantity of heat which, during unit oftime, crosses a line parallel to the base, and at a distance x fromthat base.

From equation (a) we derive also

dv 4-/T— K-^-— — (e~x siiiy — e~5x sin 3y + e~'oX sin 5y - e~7x sin 7y + &c.):

hence the integral I— K (-~\ dx, taken from x = 0, is

4/r— {(1 - e *) sin 7/ - (1 - e"8*) sin Zy + (1 - e"5") sin 5y

- (1 - e~7 ) sin 7y + &c.}.

If this quantity be subtracted from the value, which it assumeswhen x is made infinite, we find

— Ux sin y - - e"x sin Sy + - e** sin 5y - &C.) ;

and, on making y = ^?r, we have an expression for the wholequantity of heat which crosses the infinite edge (7, from thepoint whose distance from the origin is x up to the end of theplate ; namely,

SECT. IV.] PERMANENT STATE OF THE RECTANGLE. 159

which is evidently equal to half the quantity which in the sametime passes beyond the transverse line drawn on the plate ata distance x from the origin. We have already remarked thatthis result is a necessary consequence of the conditions of theproblem; if it did not hold, the part of the plate which issituated beyond the transverse line and is prolonged to infinitywould not receive through its base a quantity of heat equal tothat which it loses through its two edges; it could not thereforepreserve its state, which is contrary to hypothesis.

195. As to the expenditure of the source of heat, it is foundby supposing x = 0 in the preceding expression; hence it assumesan infinite value, the reason for which is evident if it be remarkedthat, according to hypothesis, every point of the line A has andretains the temperature 1: parallel lines which are very nearto this base have also a temperature very little different fromunity: hence, the extremities of all these lines contiguous tothe cold masses B and C communicate to them a quantity ofheat incomparably greater than if the decrease of temperaturewere continuous and imperceptible. In the first part of theplate, at the ends near to B or (7, a cataract of heat, or aninfinite flow, exists. This result ceases to hold when the distancex becomes appreciable.

196. The length of the base has been denoted by TT. If we

assign to it any value 2Z, we must write \TTJ instead of y, and

77* or

multiplying also the values of x by ^ , we must write \TT '--instead of x. Denoting by A the constant temperature of the

vbase, we must replace v by -r . These substitutions being made

in the equation (a), we have

08).

This equation represents exactly the system of permanenttemperature in an infinite rectangular prism, included betweentwo masses of ice B and C, and a constant source of heat.


197. It is easy to see either by means of this equation, orfrom Art. 171, that heat is propagated in this solid, by sepa-rating more and more from the origin, at the same time that itis directed towards the infinite faces B and G. Each sectionparallel to that of the base is traversed by a wave of heat whichis renewed at each instant with the same intensity: the intensitydiminishes as the section becomes more distant from the origin.Similar movements are effected with respect to any plane parallelto the infinite faces; each of these planes is traversed by a con-stant wave which conveys its heat to the lateral masses.

The developments contained in the preceding articles wouldbe unnecessary, if we had not to explain an entirely new theory,whose principles it is requisite to fix. With that view we addthe following remarks.

198. Each of the terms of equation (a) corresponds to onlyone particular system of temperatures, which might exist in arectangular plate heated at its end, and whose infinite edges aremaintained at a constant temperature. Thus the equationv =. e'x cos y represents the permanent temperatures, when thepoints of the base A are subject to a fixed temperature, denotedby cos y. We may now imagine the heated plate to be part of aplane which is prolonged to infinity in all directions, and denotingthe co-ordinates of any point of this plane by x and y, and thetemperature of the same point by v, we may apply to the entireplane the equation v = e~x cos y; by this means, the edges B andC receive the constant temperature 0 ; but it is not the samewith contiguous parts BB and CC; they receive and keep lowertemperatures. The base A has at every point the permanenttemperature denoted by cos y, and the contiguous parts A A havehigher temperatures. If we construct the curved surface whosevertical ordinate is equal to the permanent temperature at eachpoint of the plane, and if it be cut by a vertical plane passingthrough the line A or parallel to that line, the form of the sectionwill be that of a trigonometrical line whose ordinate representsthe infinite and periodic series of cosines. If the same curvedsurface be cut by a vertical plane parallel to the axis of xy theform of the section will through its whole length be that of alogarithmic curve.

SECT. IV.J FINAL PERMANENT STATE. 1G1

199. By this it may be seen how the analysis satisfies thetwo conditions of the hypothesis, which subjected the base to atemperature equal to cos?/, and the two sides B and G to the•temperature 0. When we express these two conditions we solvein fact the following problem: If the heated plate formed part ofan infinite plane, what must be the temperatures at all the pointsof the plane, in order that the system may be self-permanent, andthat the fixed temperatures of the infinite rectangle may be thosewhich are given by the hypothesis ?

We have supposed in the foregoing part that some externalcauses maintained the faces of the rectangular solid, one at thetemperature 1, and the two others at the temperature 0. Thiseffect may be represented in different manners; but the hypo-thesis proper to the investigation consists in regarding the prismas part of a solid all of whose dimensions are infinite, and in deter-mining the temperatures of the mass which surrounds it, so thatthe conditions relative to the surface may be always observed.

200. To ascertain the system of permanent temperatures ina rectangular plate whose extremity A is maintained at the tem-perature 1, and the two infinite edges at the temperature 0, wemight consider the changes which the temperatures undergo,from the initial state which is given, to the fixed state which isthe object of the problem. Thus the variable state of the solidwould be determined for all values of the time, and it might thenbe supposed that the value was infinite.

The method which we have followed is different, and conductsmore directly to the expression of the final state, since it isfounded on a distinctive property of that state. We now proceedto shew that the problem admits of no other solution than thatwhich we have stated. The proof follows from the followingpropositions.

201. If we give to all the points of an infinite rectangularplate temperatures expressed by equation (a), and if at the twoedges B and G we maintain the fixed temperature 0, whilst theend A is exposed to a source of heat which keeps all points of theline A at the fixed temperature 1; no change can happen in the

d2v d2vstate of the solid. In fact, the equation -^ + -y- = 0 being

(X t/s Cf.il I

F. H. n

satisfied, it is evident (Art. 170) that the quantity of heat whichdetermines the temperature of each molecule can be neitherincreased nor diminished.

The different points of the same solid having received thetemperatures expressed by equation (a) or v = cf>(oc,y), supposethat instead of maintaining the edge A at the temperature 1, thefixed temperature 0 be given to it as to the two lines B and C;the heat contained in the plate BAC will flow across the threeedges A, B, C, and by hypothesis it will not be replaced, so thatthe temperatures will diminish continually, and their final andcommon value will be zero. This result is evident since thepoints infinitely distant from the origin A have a temperatureinfinitely small from the manner in which equation (a) wasformed.

The same effect would take place in the opposite direction, ifthe system of temperatures were v = — (f) (%, y), instead of beingv = </> (%, y); that is to say, all the initial negative temperatureswould vary continually, and would tend more and more towardstheir final value 0, whilst the three edges Ay B, O preserved thetemperature 0.

202. Let v = cj> (x, y) be a given equation which expressesthe initial temperature of points in the plate BAG, whose base Ais maintained at the temperature 1, whilst the edges B and Gpreserve the temperature 0.

Let v = F(x, y) be another given equation which expressesthe initial temperature of each point of a solid plate BAC exactlythe same as the preceding, but whose three edges B, A, C aremaintained at the temperature 0.

Suppose that in the first solid the variable state which suc-ceeds to the final state is determined by the equation v = <j>(x, y, t),t denoting the time elapsed, and that the equation v = <£(#, y, t)determines the variable state of the second solid, for which theinitial temperatures are F(x, y).

Lastly, suppose a third solid like each of the two preceding:let v = / (# , y) + F(x, y) be the equation which represents itsinitial state, and let 1 be the constant temperature of the baseJ4, 0 and 0 those of the two edges B and C.

SECT. IV.] SUPERPOSITION OF EFFECTS. 163

We proceed to shew that the variable state of the third solidis determined by the equation v = </>(#, y, t) + <&(%, y, t)<

In fact, the temperature of a point m of the third solid varies,because that molecule, whose volume is denoted by M> acquiresor loses a certain quantity of heat A. The increase of tempera-ture during the instant dt is

the coefficient c denoting the specific capacity with respect tovolume. The variation of the temperature of the same point in

the first solid is --^ dt, and—rrdt in the second, the letterscM cM

d and D representing the quantity of heat positive or negativewhich the molecule acquires by virtue of the action of all theneighbouring molecules. Now it is easy to perceive that Ais equal to d -f D. For proof it is sufficient to consider thequantity of heat which the point in receives from another pointm belonging to the interior of the plate, or to the edges whichbound it.

The point m,, whose initial temperature is denoted by fv

transmits, during the instant dt, to the molecule m, a quantity ofheat expressed by q^f^ —f)dt, the factor qt representing a certainfunction of the distance between the two molecules. Thus thewhole quantity of heat acquired by m is '2ql{fl — f)dt, the sign2 expressing the sum of all the terms which would be foundby'considering the other points m2, 7??3, wA &c. which act on m;that is to say, writing q2,f2 or q3,f5, or qv fA and so on, instead ofqv fv In the same manner *Zql{F1 — F)dt will be found to bethe expression of the whole quantity of heat acquired by thesame point m of the second solid; and the factor qx is the sameas in the term 2 ^ , ( / —f)dt, since the two solids are formed ofthe same matter, and the position of the points is the same; wehave then

d^^qlA-ftdt and D-tq^-F)dt.For the same reason it will be found that

hence A = d + D and -r-, = - ^ + -r?cM CM cM

11—2

1 6 4 THEORY OF HEAT. [CHAP. III.

It follows from this that the molecule m of the third solidacquires, during the instant dt, an increase of temperature equalto the sum of the two increments which the same point wouldhave gained in the two first solids. Hence at the end of thefirst instant, the original hypothesis will again hold, since anymolecule whatever of the third solid has a temperature equalto the sum of those which it has in the two others. Thus thesame relation exists at the beginning of each instant, that is tosay, the variable state of the third solid can always be representedby the equation

203. The preceding proposition is applicable to all problemsrelative to the uniform or varied movement of heat. It shewsthat the movement can always be decomposed into several others,each of which is effected separately as if it alone existed. Thissuperposition of simple effects is one of the fundamental elementsin the theory of heat. It is expressed in the investigation, bythe very nature of the general equations, and derives its originfrom the principle of the communication of heat.

Let now v = <j> {x, y) be the equation (a) which expresses thepermanent state of the solid plate BAC, heated at its end A, andwhose edges B and C preserve the temperature 1; the initial stateof the plate is such, according to hypothesis, that all its pointshave a nul temperature, except those of the base A, whose tem-perature is 1. The initial state can then be considered as formedof two others, namely : a first, in which the initial temperatures are— <j>(x, y)y the three edges being maintained at the temperature 0,and a second state, in which the initial temperatures are + <f>{x,y),the two edges B and G preserving the temperature 0, and thebase A the temperature 1; the superposition of these two statesproduces the initial state which results from the hypothesis. Itremains then only to examine the movement of heat in each oneof the two partial states. Now, in the second, the system of tem-peratures can undergo no change; and in the first, it has beenremarked in Article 201 that the temperatures vary continually,and end with being nul. Hence the final state, properly so called,is that which is represented by v = c/> (.r, y) or equation (cc).

SECT. IV.] THE FINAL STATE IS UNIQUE. 165

If this state were formed at first it would be self-existent, andit is this property which has served to determine it for us. If thesolid plate be supposed to be in another initial state, the differ-ence between the latter state and the fixed state forms a partialstate, which imperceptibly disappears. After a considerable time,the difference has nearly vanished, and the system of fixed tem-peratures has undergone no change. Thus the variable temper-atures-converge more and more to a final state, independent ofthe primitive heating.

204. We perceive by this that the final state is unique; for,if a second state were conceived, the difference between thesecond and the first would form a partial state, which ought to beself-existent, although the edges A, B, C were maintained at thetemperature 0. Now the last effect cannot occur; similarly if wesupposed another source of heat independent of that which flowsfrom the origin A; besides, this hypothesis is not that of theproblem we have treated, in which the initial temperatures arenul. It is evident that parts very distant from the origin canonly acquire an exceedingly small temperature.

Since the final state which must be determined is unique, itfollows that the problem proposed admits no other solution thanthat which results from equation (sc). Another form may begiven to this result, but the solution can be neither extended norrestricted without rendering it inexact.

The method which we have explained in this chapter consistsin forming first very simple particular values, which agree withthe problem, and in rendering the solution more general, to theintent that v or <£ (x, y) may satisfy three conditions, namely:

It is clear that the contrary order might be followed, and thesolution obtained would necessarily be the same as the foregoing.We shall not stop over the details, which are easily supplied,when once the solution is known. We shall only give in the fol*lowing section a remarkable expression for the function <j> (or, y)whose value was developed in a convergent series in equation (a).

SECTION V.

Finite expression of the result of the solution.

205. The preceding solution might be deduced from thed2v d2v

integral of the equation -j-^ + -7-2 = 0/ which contains imaginarydec cty

quantities, under the sign of the arbitrary functions. We shallconfine ourselves here to the remark that the integral

has a manifest relation to the value of v given by the equation

— = e~x cos y — ^ e~3x cos Sy + -= e~5x cos 5y — &c.4 O D

In fact, replacing the cosines by their imaginary expressions,we have

The first series is a function of x — yj — l, and the second

series is the same function of x + yj — 1.

Comparing these series with the known development of arc tan zin functions of z its tangent, it is immediately seen that the firstis arc tan e'^vyf=r\ and the second is arc tan e~

{x+yy/=T); thusequation (a) takes the finite form

y = arc tan e - ( ^ ^ 5 + arc tan <f <-*Vzr> (B).

In this mode it conforms to the general integral

the function <j> (z) is arc tan e~% and similarly the function yjr (z).1 D. F. Gregory derived the solution from the form

V = C°S V Ix) 0 {x) +Sin {y £)Camb. Math. Journal, Vol. 1. p. 105. [A. F.]

SECT. V.] FINITE EXPRESSION OF THE SOLUTION. 167

If in equation (B) we denote the first term of the second mem-ber by p and the second by q, we have

• & , tciii q -— G ,

, x z . \ tan p + tan g 2e"* cos y 2 cos ?/whence tan (p + a) or f- -—^- = ^ = — ^ ;\x- */ 1 —tan_ptan^ 1 — e 2X e—e

whence we deduce the equation - irv = arc tan ( — _-] (0).

This is the simplest form under which the solution of theproblem can be presented.

206. This value of v or (/> (a, y) satisfies the conditions relativeto the ends of the solid, namely, <£ (x, ± \ir) = 0, and <> (0, y) = 1;

it satisfies also the general equation -j-^ + -=-a = 0, since equa-ctsc ay

tion (C) is a transformation of equation (5). Hence it representsexactly the system of permanent temperatures; and since thatstate is unique, it is impossible that there should be any othersolution, either more general or more restricted.

The equation (C) furnishes, by means of tables, the value ofone of the three unknowns v, x} y, when two of them are given; itvery clearly indicates the nature of the surface whose verticalordinate is the permanent temperature of a given point of thesolid plate. Finally, we deduce from the same equation the values

dv cl/vof the differential coefficients -y- and -y- which measure the velo-

ax aycity with which heat flows in the two orthogonal directions; andwe consequently know the value of the flow in any other direction.

These coefficients are expressed thus,d± - _ 9 ( e* + e'X \dx~ co*y\e** + 2cos2y + e'2xJ '

d^_ . ( ex- e'xdy - - z sin y

civIt may be remarked that in Article 194 the value of -y-, and

that of - j - are given by infinite series, whose sums may be easilyay

1G8 THEORY OF HEAT. [CHAP. III.

found, by replacing the trigonometrical quantities by imaginary

exponentials. We thus obtain the values of -j- and -y- which

we have just stated.

The problem which we have now dealt with is the first whichwe have solved in the,theory of heat, or rather in that part ofthe theory which requires the employment of analysis. Itfurnishes very easy numerical applications, whether we makeuse of the trigonometrical tables or convergent series, and itrepresents exactly all the circumstances of the movement ofheat. We pass on now to more general considerations.

SECTION VI.

Development of an arhitrary function in trigonometric series.

207. The problem of the propagation of heat in a rect-d2v d2v

angular solid has led to the equation -,— -f -=-^ = 0; and if itchoc cty

be supposed that all the points of one of the faces of the solidhave a common temperature, the coefficients a, h} c, d, etc. ofthe series

a cos x + b cos Sx -f c cos 5x + d cos 7x -f ... &c,must be determined so that the value of this function may beequal to a constant whenever the arc x is included between -\irand + JTT. The value of these coefficients has just been assigned;but herein we have dealt with a single case only of a more generalproblem, which consists in developing any function whatever inan infinite series of sines or cosines of multiple arcs. Thisproblem is connected with the theory of partial differentialequations, and has been attacked since the origin of that analysis.It was necessary to solve it, in order to integrate suitably theequations of the propagation of heat; we proceed to explainthe solution.

We shall examine, in the first place, the case in which it isrequired to reduce into a series of sines of multiple arcs, afunction whose development contains only odd powers of the

SECT. VI.] SERIES OF SINES OF MULTIPLE ARCS. 163

variable. Denoting such a function by <f> (x), we arrange theequation

<f>(x) = a sinx + b sin 2x -f c sin Sx + d sin 4<x + ... &c.,in which it is required to determine the value of the coefficientsa, b, c, d, &c. First we write the equation

in which <£'(0), <£""(()), $"(0), </>iv(0), &c. denote the values takenby the coefficients

dc^{x) d*$ (x) ' d^j>_{x) d44> (x)dx ' ~dx*~> d?~ ' dx* ' fJ

when we suppose x = 0 in them. Thus, representing the develop-ment according to powers of x by the equation

B + O D ^ + E

we have <f> (0) = 0, a n d <f>' (0) = A ,

$' (0)=0, f/r(0) = 5,

&c. &c.

If now we compare the preceding equation with the equation

<£> (x) = a sin # + 6 sin 2^ 4- c sin 3x + d sin 4# + e sin 5 JJ + &c,

developiog the second member with respect to powers of x, wehave the equations

A = a + 2b + 3c +U +5e + &c,

B = a + 236 + 33c + 43d + 58« + &c,

C= a + 25^ + 35c + 45cZ + 55^ + &c,

D = a + 276 + 37c + 47c? 4- 57e + &c,

These equations serve to find the coefficients a, b, c, d, e,&c, whose number is infinite. To determine them, we first re-gard the number of unknowns as finite and equal to m; thuswe suppress all the equations which follow the first m equations,


and we omit from each equation all the terms of the secondmember which follow the first m terms which we retain. Thewhole number m being given, the coefficients a, b, c, d, e, &c. havefixed values which may be found by elimination. Differentvalues would be obtained for the same quantities, if the numberof the equations and that of the unknowns were greater by oneunit. Thus the value of the coefficients varies as we increasethe number of the coefficients and of the equations which oughtto determine them. It is required to find what the limits aretowards which the values of the unknowns converge continuallyas the number of equations increases. These limits are the truevalues of the unknowns which satisfy the preceding equationswhen their number is infinite.

208. We consider then in succession the cases in which weshould have to determine one unknown by one equation, twounknowns by two equations, three unknowns by three equations,and so on to infinity.

Suppose that we denote as follows different systems of equa-tions analogous to those from which the values of the coefficientsmust be derived:

at = Alt a2 + 2b2 = A2, aa + 2b3 + 3c3 = Az,

3c4

+5e6 =A5>

&c &c

SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 171

If now we eliminate the last unknown e5 by means of thefive equations which contain A6, B6, C6, D5, E6, &c, we find

a5 (52 - 1 2 ) + 2b5 (52 - 22) + 3c5 (52 - 32) + 4da (52 - 42) = 52A5 - B5,

ab (52 - 1 2 ) + 2% (52 - 22) + S \ (52 - 32) + 4*d5 (52 - 4*) = 52i?5 - tf5,

^ (52 - 1 2 ) + 2\ (52 - 22) + 35c5 (52 - 32) + 45cZ5 (5

2 - 42) =

aB (52 - 1 2 ) + 2765 (52 - 22) + 3V6 (5

1 - 32) + 4?d5 (52 - 42) =

We could have deduced these four equations from the fourwhich form the preceding system, by substituting in the latterinstead of

and instead of A4, 52J. b — Bb>

By similar substitutions we could always pass from the casewhich corresponds to a number m of unknowns to that whichcorresponds to the number m + 1 . Writing in order all therelations between the quantities which correspond to one of thecases and those which correspond to the following case, we shallhave

-2 2 ) , c4 = cb(52-32), d^

&c. &c (c).


We have also

&c. &c (d).

From equations (c) we conclude that on representing the un-knowns, whose number is infinite, by a, b, c, d, e, &c, we musthave

a = T^T

K~ (3* - 22) (42 - 22) (52 - 22) (62 - 22) . . . '

c = (tf _ 32) (5» _ s2) (62 - 32) (72 - 32) . . . '

d = (52 - 42) (G2 - 42) (72 - 42) (82 - 4 2 ) . . . '

&a &c (e).

209. It remains then to determine the values of al9 b2> c3,d4, e5, &c.; the first is given by one equation, in which A1 enters;the second is given by two equations into which A2B2 enter; thethird is given by three equations, into which A3B3G3 enter; andso on. It follows from this that if we knew the values of

we could easily find ax by solving one equation, a2b2 by solvingtwo equations, a3b3c3 by solving three equations, and so on: afterwhich we could determine a, b, c, d, e, &c. It is required thento calculate the values of

Ai> AB*> AAC*> AACA, A5B5C5D5E5..., &c,

by means of equations (d). 1st, we find the value of A2 interms of A2 and B2\ 2nd, by two substitutions we find this valueof A1 in terms of A3B3C3; 3rd, by three substitutions we find the


same value of Ax in terms of A4B4O4D4y and so on. The successivevalues of J.1 are

Al = A£\ 32.42 - Bi (22.32 + 22.42 + 32.42) + C4 (22 + 32 + 42) - Z>4,

Al = J522.32.4?. 52 - i ? 6 (22.32.42 + 22. 32. 52 + 22. 42. 52 + 32.42. 52)

+ C, (22. 32 + 22.42 + 22.52 + 32.42+ 32.52 + 42.52)

- D 6 ( 2 2 + 3 2 + 42 + 5'2) + i;6, &c,

the law of which is readily noticed. The last of these values,which is that which we wish to determine, contains the quantitiesA, B, C, D, E, &c, with an infinite index, and these quantitiesare known; they are the same as those which enter into equa-tions (a).

Dividing the ultimate value of Al by the infinite product

22.32.42.52.62...&c,we have

+ + + + &) + G { ^ + 1 +

D \2\ 32.42 + 2\ 32. 52 + 32. 42. 52 + & a J

T42T62 + &C>

The numerical coefficients are the sums of the products whichcould be formed by different combinations of the fractions

I A 1 1 i &c

after having removed the first fraction ^ . If we represent

the respective sums of products by P19 Qv Rv S1} Tx, ... &c, andif we employ the first of equations (e) and the first of equa-tions (b), we have, to express the value of the first coefficient a,the equation

q ( 2 2 - l ) ( 3 2 - l ) ( 4 2 - l ) ( 5 2 - l ) ...23. 32 42 52...

= A - BP1 + CQ1 - DR1 + ESt - &c,

now the quantities Pv Ql9 Rt, 8l9 T^.. &c. may be easily deter-mined, as we shall see lower down; hence the first coefficient abecomes entirely known.

210. "We must pass on now to the .investigation of the follow-ing coefficients b, c, d, e, &c, which from equations (e) depend onthe quantities 62, cs, dv e6, &c. For this purpose we take upequations (b), the first has already been employed to find thevalue of al9 the two following give the value of 62, the threefollowing the value of c3, the four following the value of dv andso on.

On completing the calculation, we find by simple inspectionof the equations the following results for the values of 62, c3, dv

&c.

3c8 (I2 - 32) (22 - 32) = A51\ 22 - B3 (I

2 + 22) + C3,

4J 4 ( l 2 -4 2 ) (2 2 -4 2 ) (3 2 -4 2 )

= A^\22. 3 2 - BA (I2. 22 4-12. 32 + 22.32) + C4 (I2 + 22+ 32) - Dv

&c.

It is easy to perceive the law which these equations follow;it remains only to determine the quantities -<42/?2, J5B3GV

Afifiv &c.Now the quantities ^4.2JB2 can be expressed in terms of ASBSCS,

the latter in terms of AJifiJD^ For this purpose it suffices toeffect the substitutions indicated by equations (d); the successivechanges reduce the second members of the preceding equationsso as to contain only the ABCD, &c. with an infinite suffix,that is to say; the known quantities ABCD, &c. which enter intoequations (a) ; the coefficients become the different productswhich can be made by combining the squares of the numbers1222324252 to infinity. It need only be remarked that the firstof these squares I2 will not enter into the coefficients of thevalue of a1; that the second 22 will not enter into the coefficientsof the value of b2; that the third square 32 will be omitted onlyfrom those which serve to form the coefficients of the value of c ;and so of the rest to infinity. We have then for the values of

b<p3d4e5, &c, and consequently for those of bcde, &c, results entirelyanalogous to that which we have found above for the value ofthe first coefficient ar

211. If now we represent by P2 , Q2> i?2, S2, &c, the quantities

r + 32 42 + s*I 2 ~T~ Q 2 ' A '2 l" c a "•" ' * ' >

1 1' 1 2 >< 2 • "1 2 K2 "t" r>2 A;i "• OV5 c 2 "• ' * * ?T.3 2 12.42 12.5

V. 32. 42 I2. 32. 52 T 32.42. 52 ^ ' ' '

1 1

&c,

which are formed by combinations of the fractions p , ^ , ^ ,JL ^J o

j7,, ^ ... &c. to infinity, omitting ^ the second of these fractions

we have, to determine the value of h2, the equation1 2 __ 9 2

, - & C .

Representing in general by PnQnRn8n ... the sums of theproducts which can be made by combining all the fractions

^ > ^ > H 2 > T ^ ^ 2 - - - t o infinity, after omitting the fraction —

only; we have in general to determine the quantities aiy 62, c3>

d , £5..., &c, the following equations:

A - BPt + OQt - DBt + E8t - Ac. = Ut

&c.


212. If we consider now equations (e) winch give the valuesof the coefficients a, b, c, d, &c, we have the following results:

(22 - I2) (32 - I2) (42 - I2) (52 - I2) ...a 2 2 . 3 2 . 4 2 . 5 2 . . .

= A - BP1 + CQX - DBt + ESl - &c.,

(I2 - 2 ' )*(38-22) ( 4 2 - 2 2 ) ( 5 2 - 2 2 ) . , .Zb 12.32.42.5\. .

= A-BP2 + CQ2 - DB2 + ES2 - &c,

„ ( I 2 - 32) (22 - 32) (42 - 32) (52 - 32) . . .1*.28 .4*.5». . .

= A-BP3-CQ3-DR3 + ES3 - &c.,

, ( I 2 - 42) (22 - 42) (32 - 42) (52 - 42) . . .r . 2 2 . 3 2 . 5 2 . . .

to

= A - BP4 + C#4 - Di24 + - &c,

&c.

Remarkiog the factors which are wanting to the numeratorsand denominators to complete the double series of naturalnumbers, we see that the fraction is reduced, in the first equation

1 1 . 2 2 3 3Y • « ; in the second to - 5 . -; in the third to ^ . - ; in the

4 4fourth to - 1 . -; so that the products which multiply a, 25, 3c,

4d, &c, are alternately - and — - . It is only required then to

find the values of P&B&, P2Q2R2S2> ^3Q.A#3, &c.

To obtain them we may remark that we can make thesevalues depend upon the values of the quantities PQRST, &c,which represent the different products which may be formed

with the fractions ? ) ^ , ^ , | , J , ^ , &c, without omit-

ting any.

With respect to the latter products, their values are givenby the series for the developments of the sine. We representthen the series


1+1+!+!+!I2 + 22 3a 42 5

51 + 7& + Tii + IT* + &C.

. 1 I 1 I 1 i & e• O2 O2 * Q2 ,f 2 "• Q2 ,42 ' U C ^ '1*. 2* I 2 . 3 2 I51 .42 2 2 . 3 2 2 2 . 4 2 3 2 . 4 2

1 1 . 1 , 1 ,I 2 . 22 . 3 2 I 2 . 2 2 . 4 2 I 2 . 3 2 . 4 2 2 2 . 3 2 . 4 2

1 1 1 & c

by F, Q, R, 8, &c.

The series sin x = x - -^ + r^ — T^ + &c.

I? U! IIfurnishes the values of the quantities P, Q, R, S, &c. In fact, thevalue of the sine being expressed by the equation

we have

Whence we conclude at once that

• p = r «-]jj- *-$• s-i^-213. Suppose now that P n , Qn, BnJ 8n, &c, represent the

sums of the different products which can be made with the

fractions j - 2 , , =§ > 75 > F2 > &c-> from which the fraction —2

has been removed, w being any integer whatever; it is requiredto determine Pn, QnJ Rn, Sni &c, by means of P? Qy R, S, &c. Ifwe denote by

the products of the factors

F. H. 12

178 THEORY OF HEAT. [CHAP. IIT.

among which the factor (l - 4 ) o nty has been omitted; it follows

that on multiplying by (1 - 4 ) the quantity

1 - ? P . + 22Q» " <fK + 2*8n ~ &c,we obtain 1 - qP + q2 Q - q*R + q*8 - &c.

This comparison gives the following relations:

n

+ M5

&c;

P, = P - 1

&c,

Employing the known values of P, Q, R, 8, and making nequal to 1, 2, 3, 4, 5, &c. successively, we shall have the values ofP&R&, &c; those of PaQ2R282, &c ; those of P3QJR38a, &c.

214. From the foregoing theory it follows that the valuesof a, b, c, d, e, &c, derived from the equations

a + 2b +Sc +4td + 5e + &c. = A,a + 23b + S3c + 4?d + 53e + &c. = B,

a + 25b + 38c + 45cZ + 5Be + &c. = C,

a + 2'!b + 37c + 47d + 5'e + &c. = D,

a + 2°b + 39c + 4?d + 59e + &c. = E,

&c,

SECT. VI.] VALUES OF THE COEFFICIENTS.

are thus expressed,

- a-A2 a - A -

\7

/7T8 1 ^ Iff' U ! 1\

24 [3 ~ 2

ITT 6

32 [6 + 34 [3 37

15 + 44 |3

1 Z L|5. ~ 4e |3

&c.

179

/7T8 1 7T6 1 7T* 1 Z L 2 _ L 1 U &

Vg ~ 42 [7 + 4* |5. ~ 4e |3 + 4s; ^

215. Knowing the values of a, b, c, d, e, &c, we can substitutethem in the proposed equation

<j> {x) = a sin x + b sin 2x + c sin Sx + d sin 4x -f e sin 5# + &c,

and writing also instead of the quantities A, B, G, D, E, &c, their12—2

values £'(0), £"'(0), <£v(0), ^ ( 0 ) , <f>lx(0), &c, we have the generalequation

= sin

"(0) (j£- i ) + #'(0) (j£ - | + |

'(0) (g - I

f (0)+*""(0) (j£- i

+ &C.

We may make use of the preceding series to reduce intoa series of sines of multiple arcs any proposed function whosedevelopment contains only odd powers of the variable.

216. The first case which presents itself is that in which4>(x)=x; we find then f (0) = 1, <f>'"(0) = 0, cj>T(O) = 0, &c, and sofor the rest. We have therefore the seriea

- x = sin x — - sin 2x + ^ sin Sx — j sin 4z? + &c,A Z o- 4

which has been given by Euler.

If we suppose the proposed function to be x3, we shall have

*'(0) = 0, </>'"(0) = 13, ^ (0) = 0, ^ ( 0 ) = 0, &c,

which gives the equation

~ (*-* - | ) | sin (V -^) ^si

(A).

SECT. VI.J DEVELOPMENTS IN SERIES OF SINES. 1 8 1

We should arrive at the same result, starting from the pre-ceding equation,

-^x = sin x — -= sin 2x + ^ sin 3x — -r sin 4tx. + &c.

In fact, multiplying each member by dx, and integrating, wehave

x2 1 1 1C — -T- = cos x — ^ cos 2x + ^ c o s %x"~ 42 c o s ^^ + &c« 5

the value of the constant G is

1 7T2

a series whose sum is known to be ^ -^. Multiplying by dx theZ \6

two members of the equation

1 7T2 x2 1 1^ jTT — -y- = COS # — ^ C 0 S %X + 02 C 0 S &E — & C ,

Iand integrating we have

1 7T2X 1 X3 . 1 . « 1 • «

If now we write instead of # its value derived from theequation

- x = sin a? — ^ sin 2x + ^ sin 3# — 7 sin 4^ + &c,

we shall obtain the same equation as above, namely,

lx3 . (IT2 1 \ 1 . o /7T2 1 \ 1 . o /7T2

We could arrive in the same manner at the development inseries of multiple arcs of the powers xB, x7, x9, &c, and in generalevery function whose development contains only odd powers ofthe variable.

217. Equation (A), (Art. 216), can be put under a simplerform, which we may now indicate. We remark first, that part ofthe coefficient of sin x is the series

7T , , „ ,^v . 7T

+ p $" (0) + p #(<>) + jy V* (0) +

which represents the quantity -<ji>(7r). In fact, we have, in

general,

'(0) + £ </>" (0)

&c.

Now, the function <£(#) containing by hypothesis only oddpowers, we must have 0(0) = 0, £"(0) = 0, </>iv(0) = 0, and so on.Hence

* (a) = x<f>'(0) + 1 f "(0) + 1 ^(0) + &c.;

a second part of the coefficient of sin x is found by multiplying

by — ^ the series

&c,

whose value is - <£"(?r). We can determine in this manner the

different parts of the coefficient of sin a;, and the components ofthe coefficients of sin 2x, sin Zx, sin 4sx, &c. We may employ forthis purpose the equations:

f (0) + jg f " (0) + ^ <»v (0) + &c. = ^ (IT) ;

(0) + ^ ^ ( 0 ) + &c. = I ^-(W)

(0) + £ </>*'(()) + &c. = I (7r)

SECT. VI.] DEVELOPMENTS IN SERIES OF SINES. 183

By means of these reductions equation (A) takes the followingform:

\ 7r4>{x) = sin x ir) - I <£"(TT) + A ^(TT) - I f » + &cj

- 1 sin 2x {*(*) - i f( f f) + 1 ^ (TT) - 1 </>» + &c

- J sin 4* {<£ (TT) - i </>"(TT) + ± </>iv(7r) - i * » + &c j

+ &c. (B);

or this,

5 7r > (a?) = </> (7r) isin a; — - - sin 2a; + -^ sin 3« — &c. >

— c/)'' (TT) \ sin a? — ^3 sin 2a: + ^3 sin 3a? — &c. j-( L 6 )

+ cj>iv (7r) jsin x — ^5 sin 2x + ^ sin 3a? — &c. h

— >vi (IT) -I sin a? — ^ sin 2x + ^ sin 3x — &c. J-

+ &c. (C).

218. We can apply one or other of these formulae as often aswe have to develope a proposed function in a series of sines ofmultiple arcs. If, for example, the proposed function is ex — e~x,whose development contains only odd powers of x} we shall have

o tr—^ z^ = (s i n os — -^ sin 2x + -^ sin 3a? — &c. J

— (sin x — ^3 sin 2a? + ^3 sin 3a? — &c. J

4- (sina? — ^3 sin 2a? + ^5 sin 3a? — &c. )\ Z o /

- ( s inx - -^ sin 2x + -^ sin 3x - &c. )V

&c.

Collecting the coefficients of sin x, sin 2a?, sin 3x, sin 4#, &c,

and writing, instead of § + -5 7 + etc., its value -o—- , wen n n n n + 1

have1 (e* — e~x) sin x sin 2x sin 3# «

We might multiply these applications and derive from themseveral remarkable series. We have chosen the preceding examplebecause it appears in several problems relative to the propagationof heat.

219. Up to this point we have supposed that the functionwhose development is required in a series of sines of multiplearcs can be developed in a series arranged according to powersof the variable x, and that only odd powers enter into thatseries. We can extend the same results to any functions, evento those which are discontinuous and entirely arbitrary. To esta-blish clearly the truth of this proposition, we must follow theanalysis which furnishes the foregoing equation (B), and examinewhat is the nature of the coefficients which multiply sin a?,

sin Zx, sin 3%, &c. Denoting by - the quantity which multiplies7b

-sin nx in this equation when n is odd, and — s i n n x when n isn n neven, we have

Considering s as a function of 7r, differentiating twice, and

comparing the results, we find s +-^-y4 = <£O); an equationw air r v J ' ^

which the foregoing value of 5 must satisfy.

Now the integral of the equation 5 +— _ ? = ^ ( ) i n which sn ax ~K J

is considered to be a function of x, is

s = a cos nx + b sin nx

-f n sin nx I cos nx </> {x) dx — n cos nx I sin nx $ (x) dx.

SECT. VI.] GENERAL FORMULA. 185

If n is an integer, and the value of x is equal to 7r, we have

s — ± n Icj) (x) sin nxdx. The sign + must be chosen when n is

odd, and the sign — when that number is even. "We must makex equal to the semi-circumference IT, after the integration in-dicated; the result may be verified by developing the term

cj> (x) sin nx dx, by means of integration by parts, remarking

that the function (f> (x) contains only odd powers of the vari-able x, and taking the integral from x = 0 to x = ir.

We conclude at once that the term is equal to

I

±a

If we substitute this value of - in equation (B), taking the

sign + when the term of this equation is of odd order, and the

sign — when n is even, we shall have in general I ${x) sin nxdxfor the coefficient of sinnx; in this manner we arrive at a veryremarkable result expressed by the following equation:

9 ^#0*0 = B^nx I s^n x<l>(x) dx 4- sin 2x /sin 2#<jb (x) cfo + &c.

smtx^ (x) dx + &c (D),

the second member will always give the development requiredfor the function <f>(x)} if we integrate from x = 0 to x = 7T.1

1 Lagrange had already shewn (Miscellanea Taurinensia, Tom. in., 1766,pp. 260—1) that the function y given by the equation

y = 2 (ifV r sin XTTT AX) sin xir + 2 (i~Vr sin 2Xrir AX) sin 2#7r

+ 2 (S " Tr sin SXrir AX) sin Sxir + ... + 2 (S V r sin nXrir AX) sin nxir

receives the values r , , F3, Fg . . . ^ corresponding to the values Xly X2f Xz...Xn of

x, where Z r = - ^ - T , and A I = — r .' r n + 1 Ti + 1

Lagrange however abstained from the transition from this summation-formulato the integration-formula given by Fourier.

Cf. Riemann's Gesammelte Mathematische Werke, Leipzig, 1876, pp. 218—220of his historical criticism, Ueber die Darstellbarkeit einer Function durch eineTrigonometrische Reihe. [A. F.]

220. We see by this that the coefficients a, b, c, d, e,f, &c,which enter into the equation

5 7T(f> (x) = a sin x + b sin 2x + c sin 3o? + c? sin 4<x + &c,

and which we found formerly by way of successive eliminations,are the values of definite integrals expressed by the general term

sin ix <p (x) dx, i being the number of the term whose coefficient

is required. This remark is important, because it shews how evenentirely arbitrary functions may be developed in series of sinesof multiple arcs. In fact, if the function $ (x) be representedby the variable ordinate of any curve whatever whose abscissaextends from x — 0 to x = 7r, and if on the same part of the axisthe known trigonometric curve, whose ordinate is y = sin#, beconstructed, it is easy to represent the value of any integralterm. We must suppose that for each abscissa x} to which cor-responds one value of <£ (x)9 and one value of sin x, we multiplythe latter value by the first, and at the same point of the axisraise an ordinate equal to the product <f> (x) sin x. By this con-tinuous operation a third curve is formed, whose ordinates arethose of the trigonometric curve, reduced in proportion to theordinates of the arbitary curve which represents <f>(x). Thisdone, the area of the reduced curve taken from x = 0 to x = 7rgives the exact value of the coefficient of sin#; and whateverthe given curve may be which corresponds to <p (x), whether wecan assign to it an analytical equation, or whether it depends onno regular law, it is evident that it always serves to reducein any manner whatever the trigonometric curve; so that thearea of the reduced curve has, in all possible cases, a definitevalue, which is the value of the coefficient of sin# in the develop-ment of the function. The same is the case with the following

coefficient b, or /<£ (x) sin 2xdx.

In general, to construct the values of the coefficients a, 6, c, d, &c,we must imagine that the curves, whose equations are

y = sin Xy y = sin 2x, y = sin 3x, y = sin 4#, &c,

have been traced for the same interval on the axis of x, from

SECT. VI.] VERIFICATION OF THE FORMULA. 187

x = 0 to x = IT ; and then that we have changed these curves bymultiplying all their ordinates by the corresponding ordinates ofa curve whose equation is y = <f>(x). The equations of the re-duced curves are

y = sin x <j> (a?), y = $m2x<j>(x), y = sm3x<f>(x), &c.

The areas of the latter curves, taken from # = 0 to X — TT,are the values of the coefficients a, b, c, d, &c., in the equation

o 7r<£ (x) = a sin x + b sin 2x 4- c sin 3x + d sin 4x + &c.

221. We can verify the foregoing equation (D), (Art. 220),by determining directly the quantities a1$ a2, a3> ... ajy &c, in theequation

<f> [x) = a± sin x + a2 sin 2# + a3 sin 3ic + ... a, s i n ^ + &c.;

for this purpose, we multiply each member of the latter equationby sin ix dx, i being an integer, and take the integral from x = 0to x = 7r, whence we have

1 <f>(x) sin ix dx — aA sin x sin ix dx -\- aA sin 2# sin ix dx + &c.

+ a51 sin jo? sin ix dx + ... &c.

Now it can easily be proved, 1st, that all the integrals, whichenter into the second member, have a nul value, except only the

term at I sin ix sin ixdx; 2nd, that the value of \smixsmixdx is

^7r; whence we derive the value of ai} namely

- /<£ (x) s inisin ix dx.

The whole problem is reduced to considering the value of theintegrals which enter into the second member, and to demon-strating the two preceding propositions. The integral

2 /sin jx sin ix dx,

1 8 8 THEORY OF HEAT. [CHAP. III.

taken from x = 0 to x = 7r, in which i and j are integers, is

-.—, sin (i —j) x - -—. sin (i +j) x + C.

Since the integral must begiu when x = 0 the constant 0 isnothing, and the numbers i and j being integers, the value of theintegral will become nothing when X = TT; it follows that eachof the terms, such as

ax I sin x sin ix dx, a2 j sin 2x sin ix doc, a31 sin Sx sin i#cfc, &c,

vanishes, and that this will occur as often as the numbers i and jare different. The same is not the case when the numbers i and j

are equal, for the term -;—. sin (i —j) x to which the integral re-

duces, becomes -?, and its value is 7r. Consequently we have

2 /sin ix sin ix dx = 7r;

thus we obtain, in a very brief manner, the values of av a2, a3,...aif &c, namely,

2 r . 2 fat= - |<jf>(#) sin#cfo, a2 = - l</

2 f 2a 3 = - |<KaO s in3#d#, a4 = -

Substituting these we have

\TT$(X) = sin x j(j>(oo) sin # dx + sin 2# l > (x) sin

-f sin ia? J (-j) sin ixdx + &c.

222. The simplest case is that in which the given functionhas a constant value for all values of the variable x included

between 0 and ir; in this case the integral I sin ixdx is equal to

2%> if the number i is odd, and equal to 0 if the number i is even.

SECT. VI.] LIMITS OF THE DEVELOPMENTS. 189

Hence we deduce the equation

- 7r = sin x + Q sin 2x + •= sin 5x -f = sin 7x + &c,

which has been found before.

It must be remarked that when a function <f> (x) has been de-veloped in a series of sines of multiple arcs, the value of the series

a sin x + b sin 2x + c sin 3x + d sin 4# + &c.

is the same as that of the function </>(#) so long as the variable xis included between 0 and TT ; but this equality ceases in generalto hold good when the value of x exceeds the number 7r.

Suppose the function whose development is required to be x,we shall have, by the preceding theorem,

^ irx = sin x j x sin xdx + sin 2x I x sin 2x dx

+ sin Sx I x sin Sx dx + &c.

The integral I x sin ixdx is equal to + - ; the indices 0 and 7r,J 0 %

which are connected wi th t h e sign I , shew t h e l imi ts of t h e in t e -

gral ; t he sign + mus t be chosen when i is odd, and t h e sign —

when i is even. We have then the following equation,

•^x = sin x — Q sin 2x + « sin Sx — -r sin 4# + ^ sin 5# — &c.^ Z o 4? o

223. We can develope also in a series of sines of multiplearcs functions different from those in which only odd powers ofthe variable enter. To instance by an example which leaves nodoubt as to the possibility of this development, we select thefunction cos x, which contains only even powers of xy and whichmay be developed under the following form:

a sin x + b sin 2x + c sin 3% + d sin 4<x + e sin 5x + &c,

although in this series only odd powers of the variable enter.

We have, in fact, by the preceding theorem,

5 7r cos x = sin x I cos x sin xdx + sin 2x I cos x sin 2# rfaj

-+ sin Sx I cos x sin Sx dx + &c.

The integral I cos x sin ix dx is equal to zero when i is an

odd number, and to -^—=- when i is an even number. Supposing

successively i = 2, 4, 6, 8, etc., we have the always convergentseries

1 2 4 677T cos x = r—5 sin 2# + £-— sin 4# + -^-^ sin 6# + &c ;

or,

This result is remarkable in this respect, that it exhibits thedevelopment of the cosine in a series of functions, each one ofwhich contains only odd powers. If in the preceding equation xbe made equal to \iry we find

1 IT . 1 / 1 , 1 1 1 1 1 . \

This series is known (Introd. ad analysin. infinit. cap. x.).

224. A similar analysis may be employed for the developmentof any function whatever in a series of cosines of multiple arcs.

Let <£ (x) be the function whose development is required, wemay write

<p (x) = aQ cos Ox + ax cos x -\-a2 cos 2x + a3 cos Sx + &c.

+ aicosix+&c (m).If the two members of this equation be multiplied by cos jx,

and each of the terms of the second member integrated fromx = 0 to x = 7r; it is easily seen that the value of the integralwill be nothing, save only for the term which already containscosjx. This remark gives immediately the coefficient < ; it issufficient in general to consider the value of the integral

Icosjx cos ix dx,

SECT. VI.J DEVELOPMENT IN SERIES OF COSINES. 191

taken from x — 0 to x = IT, supposing j and i to be integers. Wehave

f 1 . . 1J cos jx cos ix dx = . .. sin (j + i) x -f . _ sin (^ - i) x 4- c.

This integral, taken from x = 0 to X — TT, evidently vanisheswhenever J and i are two different numbers. The same is notthe case when the two numbers are equal. The last term

1 . , . -. - sin (j — i) x

becomes - , and its value is ^IT, when the arc x is equal to TT.

If then we multiply the two terms of the preceding equation (m)by cos ia?, and integrate it from 0 to 7r, we have

l<fl<f) (x) COS ix dx = ^TTdi,

an equation which exhibits the value of the coefficient a$.

To find the first coefficient a0, it may be remarked that inthe integral

if j = 0 and i — 0 each of the terms becomes ^ , and the value

of each term is \ir; thus the integral I cos jx cos ix dx taken

from x = 0 to x = ir is nothing when the two integers j and iare different: it is \ir when the two numbers j and i are equalbut different from zero; it is equal to IT when j and i are eachequal to zero; thus we obtain the following equation,

1 fv T71* f77

^7r</>(#) = ^ I <f>(x)dx+cosx I <£(s£)cosa?cfoc+cos2# </>WO 70 JO

+ cos Sx I 6 (a?) cos Sx dx + &c. (?i) \Jo

1 The process analogous to ( l) in Art. 222 fails here; yet we see, Art. 177, thatan analogous result exists. [E. h. E.]

This and the preceding theorem suit all possible functions,whether their character can be expressed by known methods ofanalysis, or whether they correspond to curves traced arbitrarily.

225. If the proposed function whose development is requiredin cosines of multiple arcs is the variable x itself; we may writedown the equation

^irx = a0 + ax cos x + a2 cos 2x + a3 cos Sx + ... + a% cos ix + &c,

and we have, to determine any coefficient whatever ai9 the equa-

tion a{= I x cos ix dx. This integral has a nul value when i isJ o

2an even number, and is equal to — - when i. is odd. We have at

%

the same time a0 = j 7r2. We thus form the following series,

1 A cos x . cos 3x . cos 5x A cos 7x 0

2 7T 3V 527T 7 7T

We may here remark that we have arrived at three differentdevelopments for x, namely,

1 . 1 • « 1 • o ! • . 1 • e JP- x = s m a; — ^ s m 2 x + ^ s m 0 ^ — 7 sin 45? + - s m 5a? — &a,j ^ <5 4f o

1 2 . 2 . 2- x = - sin # — -ZT- sin 3a; + •—- sin 5x — &c. (Art. 181),L IT O IT 5 7T

1 1 2 2 o 2 K p

r 0; = -j 7T COS # — ^o— COS 3x — r^— COS OX — &C.2 4 7T 3 V 5 7T

It must be remarked that these three values of \x ought notto be considered as equal; with reference to all possible values ofx, the three preceding developments have a common value onlywhen the variable x is included between 0 and |7r. The con-struction of the values of these three series, and the comparison ofthe lines whose ordinates are expressed by them, render sensiblethe alternate coincidence and divergence of values of thesefunctions.

To give a second example of the development of a function ina series of cosines of multiple arcs, we choose the function sin x,

SECT. VI.J TRIGONOMETRICAL DEVELOPMENTS. 193

which contains only odd powers of the variable, and we may sup-pose it to be developed in the form

a + b cos x + c cos 2x + d cos 3x + &c.

Applying the general equation to this particular case, we find,as the equation required,

I . 1 cos 2% cos 4# cos 6x ~-7T Sin * = g - -y-g- - -g-g g - y - &C.

Thus we arrive at the development of a function which con-tains only odd powers in a series of cosines in which only evenpowers of the variable enter. If we give to x the particular value^7r, we find

TC & J L . O O . t ? t / . f i , %y

Now, from the known equation,

1 _ 1 1 1 1__1_i 7 3 " " 3 + 5 7 + 9 I I 4 '

we derive

8 ™ = 173 + ^TT + 9 .11 + 13 . 15 + '

and alsoi i i i i

- - — &c.8 2 3 .5 7 . 9 11.13

Adding these two results we have, as above,

1 1 1 1 1 1 14 ^ - 2 + 173 " 375 + 577 ~ 779 + STTl " ^C'

226. The foregoing analysis giving the means of developingany function whatever in a series of sines or cosines of multiplearcs, we can easily apply it to the case in which the function to bedeveloped has definite values when the variable is includedbetween certain limits and has real values, or when the variable isincluded between other limits. We stop to examine this particularcase, since it is presented in physical questions which depend onpartial differential equations, and was proposed formerly as an ex-ample of functions which cannot be developed in sines or cosines

F. H. 13

of multiple arcs. Suppose then that we have reduced to a series ofthis form a function whose value is constant, when x is includedbetween 0 and a, and all of whose values are nul when x is in-cluded between a and 7r. We shall employ the general equation(Z>), in which the integrals must be taken from x = 0 to X = TT.The values of <j>(x) which enter under the integral sign beingnothing from x = a to x = 7r, it is sufficient to integrate from x = 0to x = a. This done, we find, for the series required, denoting byh the constant value of the function,

1 f / x 7 (,.. N • 1 — cos 2a~7n£(#) = h Ml — cos a) sin x H ^ sin zx

1 - cos 3a . p }

If we make 1I = \TT} and represent the versed sine of the arc xby versin x, we have

<j> (x) = versin a sin # + ^ versin 2a sin 2x + ^ versin 3^ sin 3# + &C.1

This series, always convergent, is such that if we give to x anyvalue whatever included between 0 and a, the sum of its termswill be |-7r; but if we give to x any value whatever greater thana and less than ^iry the sum of the terms will be nothing.

In the following example, which is not less remarkable, the

values of (f> (x) are equal to sin — for all values of x included

between 0 and a, and nul for values of x between a and ir. Tofind what series satisfies this condition, we shall employ equa-tion (Z>).

The integrals must be taken from x = 0 to x = TT ; but it issufficient, in the case in question, to take these integrals fromx=0 to x=a, since the values of </>(#) are supposed nul in therest of the interval. Hence we find

o (sin a sin x sin 2a sin 2x sin 3a sin Sx 0

1 In what cases a function, arbitrary between certain limits, can be developedin a series of cosines, and in what cases in a series of sines, has been shewn bySir W. Thomson, Camb. Math. Journal, Vol. n. pp. 258—262, in an articlesigned P. Q. B., On Fourier's Expansions of Functions in Trigonometrical Series.

[A. F.]

SECT. VI.] TRIGONOMETRICAL DEVELOPMENTS. 195

If a be supposed equal to ir, all the terms of the series vanish,

except the first, which becomes - , and whose value is sin x; we

have then <f>(x) = since.

227. The same analysis could be extended to the case inwhich the ordinate represented by <p (x) was that of a line com-posed of different parts, some of which might be arcs of curvesand others straight lines. For example, let the value of the func-tion, whose development is required in a series of bosines of

/7T\

multiple arcs, be Ur) — x2, from # = 0 to cc = |7r, and be nothingfrom x = \TT to x = IT. We shall employ the general equation (n),and effecting the integrations within the given limits, we find

that the general term1 I ( ^ ) — x2 cos ixdx is equal to -z when ij \_y^j j ^

is even, to — - when i is the double of an odd number, and to

— —z when i is four times an odd number. On the other hand, we1 7T3 1 f

find - ^ for the value of the first term ~ \<fr(x)dx. We have then

the following development:cos Sx cos 5x cos 7x n

+ + + &

cos 2x cos 4>x cos 6x „9^ 42 (\2

The second member is represented by a lino composed of para-bolic arcs and straight lines.ig

228. In the same manner we can find the development of afunction of x which expresses the ordinate of the contour of atrapezium. Suppose <f> Qc) to be equal to x from x = 0 to x = c,that the function is equal to a from X~OL to X = IT— a, and lastlyequal to TT — x, from x = IT — a to x = TT. TO reduce it to a series

/7r\2sini;» xn . . 2' . sin ?se« = ( — ) : r S i n IX - ~2 X COS 105 + 2 ^ .

[R. L. E.]

13—2

of sines of multiple arcs, we employ the general equation (D).

The general term /<£ (x) sin ix dx is composed of three different

2parts, and we have, after the reductions, -^sin m for the coefficient

of sin ix, when i is an odd number; but the coefficient vanisheswhen i is an even number. Thus we arrive at the equation

7zird> (x) = 2 -Urn a sin x + -2 sin 3a sin Soo + ^ sin 5a sin 5#[ #3

2 s i n 7a sin 7a; + &c. J- (X).1

If we supposed a = ^7r, the trapezium would coincide with anisosceles triangle, and we should have, as above, for the equa-tion of the contour of this triangle,

-7r6(oo) = 2 (sinx — ^ s i n 3%-f -g sin 5 # - ^ s i n 7# + &c.h2

a series which is always convergent whatever be the value of x.In general, the trigonometric series at which we have arrived,in developing different functions are always convergent, but ithas not appeared to us necessary to demonstrate this here; for theterms which compose these series are only the coefficients of termsof series which give the values of the temperature; and thesecoefficients are affected by certain exponential quantities whichdecrease very rapidly, so that the final series are very convergent.With regard to those in which only the sines and cosines ofmultiple arcs enter, it is equally easy to prove that they areconvergent, although they represent the ordinates of discontinuouslines. This does not result solely from the fact that the valuesof the terms diminish continually; for this condition is notsufficient to establish the convergence of a series. It is necessarythat the values at which we arrive on increasing continually thenumber of terms, should approach more and more a fixed limit,

1 The accuracy of this and other series given by Fourier is maintained bySir W. Thomson in the article quoted in the note, p. 194!

2 Expressed in cosines between the limits 0 and 7r,

ITT(P (x)=^ - (cos 2ic+ gT, cos 6x+-2 cos 10# + &c. J .

Cf. De Morgan's Biff, and Int. Calc, p. 622. [A. F.]

SECT. VI.] GEOMETRICAL ILLUSTRATION. 197

and should differ from it only by a quantity which becomes lessthan any given magnitude: this limit is the value of the series.Now we may prove rigorously that the series in question satisfythe last condition.

229. Take the preceding equation (\) in which we can giveto x any value whatever; we shall consider this quantity as anew ordinate, which gives rise to the following construction.

Fig. 8.

mkm 1

Having traced on the plane of x and y (see fig. 8) a rectanglewhose base Oir is equal to the semi-circumference, and whoseheight is \ir \ on the middle point m of the side parallel to thebase, let us raise perpendicularly to the plane of the rectanglea line equal to \iry and from the upper end of this line drawstraight lines to the four corners of the rectangle. Thus will beformed a quadrangular pyramid. If we now measure from thepoint 0 on the shorter side of the rectangle, any line equal to a,and through the end of this line draw a plane parallel to the baseOir, and perpendicular to the plane of the rectangle, the sectioncommon to this plane and to the solid will be the trapezium whoseheight is equal to a. The variable ordinate of the contour ofthis trapezium is equal, as we have just seen, to

4 / 1 1 . . \— [sin a sinx + =i s i f l 3a sin %x + —2 sin 5a sin 5x + &c. .IT \ O O J

It follows from this that calling x, y, z the co-ordinates of anypoint whatever of the upper surface of the quadrangular pyramidwhich we have formed, we have for the equation of the surfaceof the polyhedron, between the limits

1 sin x sin y sin Sx sin Sy sin5^sin5z/2™= l ' + ¥ + gff—-

This convergent series gives always the value of the ordinatez or the distance of any point whatever of the surface from theplane of x and y.

The series formed of sines or cosines of multiple arcs aretherefore adapted to represent, between definite limits, all possiblefunctions, and the ordinates of lines or surfaces whose form isdiscontinuous. Not only has the possibility of these develop-ments been demonstrated, but it is easy to calculate the termsof the series; the value of any coefficient whatever in theequation

 (x)y or the form of the curvewhich it represents, the integral has a definite value which maybe introduced into the formula. The values of these definite

integrals are analogous to that of the whole area |<£ (x) dx in-cluded between the curve and the axis in a given interval, or tothe values of mechanical quantities, such as the ordinates of thecentre of gravity of this area or of any solid whatever. It isevident that all these quantities have assignable values, whetherthe figure of the bodies be regular, or whether we give to theman entirely arbitrary form.

230. If we apply these principles to the problem of the motionof vibrating strings, we can solve difficulties which first appearedin the researches of Daniel Bernoulli. The solution given by thisgeometrician assumes that any function whatever may always bedeveloped in a series of sines or cosines of multiple arcs. Nowthe most complete of all the proofs of this proposition is thatwhich consists in actually resolving a given function into such aseries with determined coefficients.

In researches to which partial differential equations are ap-plied, it is often easy to find solutions whose sum composes amore general integral; but the employment of these integralsrequires us to determine their extent, and to be able to dis-

SECT. VI.] REMARKS ON THE DEVELOPMENTS. 199

tinguish clearly the cases in which they represent the generalintegral from those in which they include only a part. It isnecessary above all to- assign the values of the constants, andthe difficulty of the application consists in the discovery of thecoefficients. It is remarkable that we can express by convergentseries, and, as we shall see in the sequel, by definite integrals,the ordinates of lines and surfaces which are not subject to acontinuous law1. We see by this that we must admit into analysisfanctions which have equal values, whenever the variable receivesany values whatever included between two given limits, eventhough on substituting in these two functions, instead of thevariable, a number included in another interval, the results ofthe two substitutions are not the same. The functions whichenjoy this property are represented by different lines, whichcoincide in a definite portion only of their course, and offer asingular species of finite osculation. These considerations arisein the calculus of partial differential equations; they throw a newlight on this calculus, and serve to facilitate its employment inphysical theories.

231. The two general equations which express the develop-ment of any function whatever, in cosines or sines of multiplearcs, give rise to several remarks which explain the true meaningof these theorems, and direct the application of them.

If in the series

a + b cos x + c cos 2x + d cos 3x + e cos 4<x + &c,

we make the value of x negative, the series remains the same; italso preserves its value if we augment the variable by any multiplewhatever of the circumference 2TT. Thus in the equation

1 1 f f- TT<\> (x) = 9 I <£ (#) dx + cos x I </> (x) cos xdx

+ cos 2x j(j) (x) cos 2xdx + cos 3x /</> (x) cos3xdx + &c....(V),

the function <£ is periodic, and is represented by a curve composedof a multitude of equal arcs, each of which corresponds to an

1 Demonstrations have been supplied by Poisson, Deflers, Dirichlet, Dirksen,Bessel, Hamilton, Boole, De Morgan, Stokes. See note, pp. 208, 209. [A. F.]

interval equal to 2TT on the axis of the abscissae. Further, each ofthese arcs is composed of two symmetrical branches, which cor-respond to the halves of the interval equal to 2?r.

Suppose then that we trace a line of any form whatever <£<£a(see fig. 9.), which corresponds to an interval equal to 7r.

Fig. 9.

If a series be required of the form

a + b cos x + ccos 2x + d cos Sx + &c.;

such that, substituting for x any value whatever X included be-tween 0 and 7r, we find for the value of the series that of theordinate Xcj), it is easy to solve the problem: for the coefficientsgiven by the equation (v) are

1 f 2 f 2 f- \<f> (%) dx, — /<£ [x) cos 2xdx, - /<£ (x) cos Sxdx, &c.

These integrals, which are taken from x = 0 to x = 7r, havingalways measurable values like that of the area Ocfyzir, and theseries formed by these coefficients being always convergent, thereis no form of the line (jxpz, for which the ordinate Xcj> is notexactly represented by the development

a -f b cos x + c cos 2x + d cos Sx + e cos 4<x + &c.

The arc <£<£a is entirely arbitrary; but the same is not the casewith other parts of the line, they are, on the contrary, determinate;thus the arc cf>x which corresponds to the interval from 0 to — IT isthe same as the arc </><z; and the whole arc a<f>a is repeated onconsecutive parts of the axis, whose length is 27r.

We may vary the limits of the integrals in equation (y). Ifthey are taken from x = — ir to x=-it the result will be doubled:it would also be doubled if the limits of the integrals were0 and 27r; instead of being 0 and TT. We denote in general by the

SECT. VI.J GEOMETRICAL ILLUSTRATION. 201

sign I an integral which begins when the variable is equal to a,J a

and is completed when the variable is equal to b; and we writeequation (n) under the following form :

^7T(j)(x) = ^ I <f)(x)dx+cosx\ <p(x)cosxdx + cos2xl <})(x)cos2xdx

+ cos Sx I $ (x) cos Sxdx + etc (v).J o

Instead of taking the integrals from x = 0 to x = IT, we mighttake them from x = 0 to x = 2TT, or from x = — IT to x = 7r; but ineach of these two cases, ir<\> [x) must be written instead of 7r<£ (a?)in the first member of the equation.

232. In the equation which gives the development of anyfunction whatever in sines of multiple arcs, the series changessign and retains the same absolute value when the variable xbecomes negative; it retains its value and its sign when thevariable is increased or diminished by any multiple whatever of

Fig. 10.

the circumference 2TT. The arc <jxf>a (see fig. 10), which cor-responds to the interval from 0 to IT is arbitrary; all the otherparts of the line are determinate. The arc (fxjxx, which correspondsto the interval from 0 to - TT, has the same form as the given arc<pcj)a; but it is in the opposite position. The whole arc aKfxfxfx^a isrepeated in the interval from IT to 37r, and in all similar intervals.We write this equation as follows:

- TT6 (X) = sin x j <£ (x) sin xdx + sin 2x I <f> (x) sin 2xdx2 Jo Jo

+ sin 3x I 6 (x) sin Sxdx + &c (ji).JO

We might change the limits of the integrals and write

I or I instead of I ;

but in each of these two cases it would be necessary to substitutein the first member TT<£ (X) for \TT^> (X).

233. The function cj> (x) developed in cosines of multiple arcs,is represented by a line formed of two equal arcs placed sym-

Fig. 11.

9

F

-7T

/

V

\fIP

metrically on each side of the axis of y, in the interval from— 7r to +7r (see fig. 11) ; this condition is expressed thus,

<£(#) = <£(-#).The line which represents the function ty (x) is, on the contrary,formed in the same interval of two opposed arcs, which is what isexpressed by the equation

Any function whatever F(x), represented by a line tracedarbitrarily in the interval from — ir to + IT, may always be dividedinto two functions such as <jf> (x) and yfr (x). In fact, if the lineF'F'mFF represents the function F{x), and we raise at the pointo the ordinate om, we can draw through the point m to the rightof the axis om the arc mff similar to the arc mFrF of the givencurve, and to the left of the same axis we may trace the arc mffsimilar to the arc mFF; we must then draw through the point ma line (/>'</>'ra<£<£ which shall divide into two equal parts the differ-ence between each ordinate xF or x f and the corresponding

SECT. VI.] GEOMETRICAL DEMONSTRATION. 203

ordinate xf or x'Fr. We must draw also the line ^ ' ^ ' 0 ^ ^ whoseordinate measures the half-difference between the ordinate ofF'F'mFF and that of ff'mff. This done the ordinate of thelines F'F'mFF, and f'f'mffhemg denoted by F (x) and f{x)respectively, we evidently have f(x) = F(— x); denoting also theordinate of </>'(/>'m</><£ by <£ (#), and that of i/Ajr'O-vJn/r by ty (a?),we have

F(x) = cf>(x)+f{x) and /(a?) = <j> {x) - f (x) = .F(- »),

hence

</,(^) = ^ ( ^ ) + | J P ( - a ; ) and f (X) = \F{x)-\F{-X),

whence we conclude that

(j> (x) = <p (— x) and ^ (a?) = — ty (— a?),

which the construction makes otherwise evident.

Thus the two functions <£ (a?) and ^ (x), whose sum is equal toF (x) may be developed, one in cosines of multiple arcs, and theother in sines.

If to the first function we apply equation (i/), and to the secondthe equation (/x), taking the integrals in each case from x = — irto x = 7r, and adding the two results, we have

— & I$0*0 d% + cos x l(f)(x) cos xdx + cos 2x \<$>{%) cos 2xdx + &c.

+ sin x \^r(x) sin xdx + sin 2x j^r(x) sin 2xdx + &c.

The integrals must be taken from x = — ir to x — ir. It may now

be remarked, that in the integral I </> (x) cos x dx we could,J -IT

without changing its value, write <f>(x) + yjr(x) instead of cf>(x):for the function coscc being composed, to right and left of theaxis of xf of two similar parts, and the function ^ (x) being, on the

contrary, formed of two opposite parts, the integral I ty(x) cos xdx

vanishes. The same would be the case if we wrote cos 2x orcos Sx, and in general cos ix instead of cos x, i being any integer

r+ir

from 0 to infinity. Thus the integral I <f>(x) cos ixdx is the sameJ - 7 T

as the integralr+7r f+7r

I bfr 0*0 + 0*0] cos x dx, or I .F(#) COS ix dx.J "IT J -IF

r+ir

It is evident also that the integral I ^ (x) sin ix dx is equal

to the integral I .F(#) sin ix dx, since the integral I <£ (a?) sin i#cfo?J -IT J -It

vanishes. Thus we obtain the following equation (p), which servesto develope any function whatever in a series formed of sines andcosines of multiple arcs :

(p)

+ cos x I F(x) cos x dx + cos 2x I F(x) cos 2x dx + &c.

+ sin x \F(x) sin x dx + sin 2x \F(x) sin 2x dx + &c.

234. The function F(x), which enters into this equation, isrepresented by a line F'F'FF, of any form whatever. The arcF'F'FF, which corresponds to the interval from —IT to +7r, isarbitrary; all the other parts of the line are determinate, and thearc F'F'FF is repeated in each consecutive interval whose lengthis 2TT. We shall make frequent applications of this theorem, andof the preceding equations (//,) and (v).

If it be supposed that the function F(x) in equation (p) is re-presented, in the interval from — IT to + IT, by a line composed oftwo equal arcs symmetrically placed, all the terms which containsines vanish, and we find equation (y). If, on the contrary, theline which represents the given function F(x) is formed of twoequal arcs opposed in position, all the terms which do not containsines disappear, and we find equation (/*). Submitting the func-tion F{x) to other conditions, we find other results.

If in the general equation (p) we write, instead of the variable

x, the quantity —, x denoting another variable, and 2r the length

SECT. VI.J MODIFICATION OF THE SERIES. 205

of the interval which includes the arc which represents F(x);

the function becomes F(™\ which we may denote by f(x).

The limits x = — ir and x = IT become — = — 7r, •— = IT ; wer r

have therefore, after the substitution,

(P).

X f J*/ \ TTX -. 2lTX f j . , x 2TTX 7

+ cos 7r - fix) cos — dx + cos /(a?) cos ax -f- etc.X [ „, , . 7TX * . 2lTX f ». N . 27T# 7

+ siU7r- /(ar) sin — ao; + sin / (#) sm t& + etc.

All the integrals must be taken like the first from x = — r tox = + r. If the same substitution be made in the equations (y)and (fi), we have

^ dx + cos T'f N 2-rrxc o s / ()f » . N 2-rrx 7 , 0 /AT./ / (x) cos dx + &c (JN),

and

= sin —

-^- If (x) si ^ + &c (M).

In the first equation (P) the integrals might be taken fromfrom x — 0 to x = 2r, and representing by x the whole interval 2r,we should have *

1 It has been shewn by Mr J. O'Kinealy that if the values of the arbitraryfunction/(ic) be imagined to recur for every range of x over successive intervals \we have the symbolical equation

and the roots of the auxiliary equation being

9 w = o, 1, 2, 3 . . . co, [Turnover.

+ cos ^ff(x) cos *gdx + cos ^ ]><*) cos 5 ? cfe + &c.

. 2-7TCC f / . , x . 2 7 r # 7 . 4 < 7 r # /"/./ v . 4 7 r # ,+ sin -^r- / (a?) sin - ^ - a# + sin - y - 1/ W sin - ^ - cte + &c.

235. It follows from that which has been proved in this sec-tion, concerning the development of functions in trigonometricalseries, that if a function f(x) be proposed, whose value in a de-finite interval from x = 0 to x = X is represented by the ordinateof a curved line arbitrarily drawn; we can always develope thisfunction in a series which contains only sines or only cosines, orthe sines and cosines of multiple arcs, or the cosines only of oddmultiples. To ascertain the terms of these series we must employequations (M), (N), (P).

The fundamental problems of the theory of heat cannot becompletely solved, without reducing to this form the functionswhich represent the initial state of the temperatures.

These trigonometric series, arranged according to cosines orsines of multiples of arcs, belong to elementary analysis, like theseries whose terms contain the successive powers of the variable.The coefficients of the trigonometric series are definite areas, andthose of the series of powers are functions given by differentiation,in which, moreover, we assign to the variable a definite value. Wecould have added several remarks concerning the use and pro-perties of trigonometrical series; but we shall limit ourselves toenunciating briefly those which have the most direct relation tothe theory with which we are concerned.

it follows that

/(*) =4, + ^ cos ^ + J2cos2 2 ~ + A3 cos 3 2~ + &G.A A X

A A \

The coefficients being determined in Fourier's manner by multiplying both. , , cos 2irx _ .

sides by g.Q n — and integrating from 0 to A. (Philosophical Magazine, August

1874, pp. 95, 9G). [A. F.]

SECT. VI.] REMARKS ON THE SERIES. 207

1st. The series arranged according to sines or cosines of mul-tiple arcs are always convergent; that is to say, on giving to thevariable any value whatever that is not imaginary, the sum of theterms converges more and more to a single fixed limit, which isthe value of the developed function.

2nd. If we have the expression of a function f(x) which cor-responds to a given series

a + b cos x + c cos 2x + d cos Sx + e cos 4*x + &c,

and that of another function </> (as), whose given development is

a + yS cos x + 7 cos 2x + 8 cos Sx + € cos 4<x + &c,

it is easy to find in real terms the sum of the compound series

ad + bfl + cy + dS + ee + &C,1

and more generally that of the series

aoi + bj3 cos x + cy cos 2x + dS cos Sx + ee cos 4<x + &c,which is formed by comparing term by term the two given series.This remark applies to any number of series.

3rd. The series (P) (Art. 234) which gives the developmentof a function F (x) in a series of sines and cosines of multiple arcs,may be arranged under the form

+ cos x I F(a) cos ada + cos 2x I F (a) cos 2odi + &c.

in x I F (a) sin adz + sin %x I F (a) sin 2adz + &c.sin

a being a new variable which disappears after the integrations.

We have then

+ cos x cos a + cos 2x cos 2a + cos Sx cos 3a + &c.

+ sin x sin a + sin 2^ sin 2a + sin Sx sin 3a + &c. J-,

1 We shall have

}. [E. L. E.]


or

F(x) = l r F(a) doL | | + cos (x - a) + cos 2 (x - a) + &c j .

Hence, denoting the sum of the preceding series by

2 cos % (x — a)

taken from i = 1 to i = oo, we have

F{x) =^JF (a) doi | | + t cos i(x - a)I -

The expression - + 2 cos i (x — a) represents a function of x

and a, such that if it be multiplied by any function whatever F(a),and integrated with respect to a between the limits a = — ir anda = 7r, the proposed function F(a) becomes changed into a likefunction of x multiplied by the semi-circumference ir. It will beseen in the sequel what is the nature of the quantities, such as

- + 2cos/(a?—a), which enjoy the property we have just enun-A

ciated.

4th. If in the equations (M), (N), and (P) (Art 234), whichon being divided by r give the development of a function /(#),we suppose the interval r to become infinitely large, each term ofthe series is an infinitely small element of an integral; the sum ofthe series is then represented by a definite integral. When thebodies have determinate dimensions, the arbitrary functions whichrepresent the initial temperatures, and which enter into the in-tegrals of the partial differential equations, ought to be developedin series analogous to those of the equations (M), (N), (P) ; butthese functions take the form of definite integrals, when thedimensions of the bodies are not determinate, as will be ex-plained in the course of this work, in treating of the free diffusionof heat (Chapter ix.).

on Section VI. On the subject of the development of a function whosevalues are arbitrarily assigned between certain limits, in series of sines andcosines of multiple arcs, and on questions connected with the values of suchseries at the limits, on the convergeney of the series, and on the discontinuityof their values, the principal authorities are

Poisson, Tiieorie mathematique cle la Chaleur, Paris, 1835, Chap. vn. Arts.92—102, Sur la maniere d'exprimer les fonctions arbitraires par des series de

SECT. VII.J LITERATURE. 209

quantith periodiques. Or, more briefly, in his Traits de Mecanique, Arts. 325—328.Poisson's original memoirs on the subject were published in the Journal de VEcolePoly technique, Cahier 18, pp. 417—489, year 1820, and Cahier 19, pp. 404—509,year 1823.

De Morgan, Differential and Integral Calculus. London, 1842, pp. 609—617.The proofs of the developments appear to be original. In the verification of thedevelopments the author follows Poisson's methods.

Stokes, Cambridge Philosophical Transactions, 1847, Vol. vm. pp. 533—556.On the Critical values of the sums of Periodic Series. Section I. Mode of ascertain-ing the nature of the discontinuity of a function ivhich is expanded in a seriesof sines or cosines, and of obtaining the developments of the derived functions.Graphically illustrated.

Thomson and Tait, Natural Philosophy, Oxford, 1867, Vol. i. Arts. 75—77.Donkin, Acoustics, Oxford, 1870, Arts. 72—79, and Appendix to Chap. iv.Matthieu, Cours de Physique Mathematique, Paris,, 1873, pp. 33—36.Entirely different methods of discussion, not involving the introduction of

arbitrary multipliers to the successive terms of the series were originated byDirichlet, Crelle's Journal, Berlin, 1829, Band iv. pp. 157—169. Sur la con-

vergence des series trigonometriques qui servent a representer une fonction arbitraireentre les limites donnees. The methods of this memoir thoroughly deserve at-tentive study, but are not yet to be found in English text-books, Another memoir,of greater length, by the same author appeared in Dove's Repertorium der Physik,Berlin, 1837, Baud i. pp. 152—174. Ueber die Darstellwig ganz ivillkuhrlicherFunctionen durch Sinus- und Cosinusreihen. Von G. Lejeune Dirichlet.

Other methods are given byDirksen, Crelle's Journal, 1829, Band iv. pp. 170—178. Ueber die Convergenz

einer nach den Sinussen nnd Cosinussen der Yielfachen eines Winkels forUchreiten-den Reihe.

Bessel, Astronomische Nachrichten, Altona, 1839, pp. 230—238. Ueber denAusdruck einer Function <p (x) durch Cosinusse und Sinusse der Vielfachen von x.

The writings of the last three authors are criticised by Riemann, GesammelteMathematische Werke, Leipzig, 1876, pp. 221—225. Ueber die Darstellbarkeit einerFunction durch eine Trigonometrische Reihe.

On Fluctuating Functions and their properties, a memoir was published bySir W. R. Hamilton, Transactions of the Royal Irish Academy, 1843, Yol. xix. pp.264—321. The introductory and concluding remarks may at this stage be studied.

The writings of Deflers, Boole, and others, on the subject of the expansionof an arbitrary function by means of a double integral (Fourier's Theorem) willbe alluded to in the notes on Chap. IX. Arts. 361, 362. [A. F.J

SECTION VII.

Application to the actual problem.

236. We can now solve in a general manner the problem ofthe propagation of heat in a rectangular plate BA C, whose end Ais constantly heated, whilst its two infinite edges B and G aremaintained at the temperature 0.

F. H. 14

2 1 0 THEORY OF HEAT. [CHAP. ITT.

Suppose the initial temperature at all points of the slab BACto be nothing, but that the temperature at each point m of theedge A is preserved by some external cause, and that its fixedvalue is a function f(x) of the distance of the point m from theend 0 of the edge A whose whole length is 2r; let v be theconstant temperature of the point m whose co-ordinates are x andy, it is required to determine v as a function of x and y.

The value v = ae~my sin mx satisfies the equation

a and m being any quantities whatever. If we take m = i~ ,

i being an integer, the value ae ™r sin vanishes, when x = r,

whatever the value of y may be. We shall therefore assume, as amore general value of v,

-TTI . <TTX -2n- . 2TTX -Zn~ . STTX O

v = a1e r sin \-a.e r sin h ax r sin h &c.

If y be supposed nothing, the value of v will by hypothesisbe equal to the known function f(x). We then have

/•/ \ • TTX . 2777£ . S7TX n

fix) = ax sin — + a2 sm h a3 sm h &c.

The coefficients av a2, av &c. can be determined by means ofequation (M), and on substituting them in the value of v we have

\rv =

+ eSnr sin ^ / / ( * ) sin ?^«fa + &c.

237. Assuming r = TT in the preceding equation, we have thesolution under a more simple form, namely

2 Try = e"* sin * J / (a) sin atfa + e"* sin 2« Jf(a) sin

+ e'3!/ sin Sx \f(x) sin Sa ;^ + &c (a),

SECT. VII.] APPLICATION OF THE THEORY. 211

or

i r2 7rv = I / (*) d<* ifv sin x sin a + e~22/ sin 2# sin 2a

+ ef3" sin 3x sin 3a + &c.)

a is a new variable, which disappears after integration.

If the sum of the series be determined, and if it be substitutedin the last equation, we have the value of v in a finite form. Thedouble of the series is equal to

e*[cos (x — a) — cos (x + a)] + e'2y [cos2 (x — a) - cos 2 (x + a)]

+ e'zy [cos 3 (x - a) - cos 3 (x + a)] + &c.;

denoting by F (y,p) the sum of the infinite series

e~y cosp + e~2y cos 2p -h e~3y cos 3p + &c,

we find

o

We have also

= K + e + e + &c.l

whence

cos (x — a) — e~7TV •

or

2 (gy - e"y) sin a? sin «f(a) di) —

or, decomposing the coefficient into two fractions,

i-OL) + e"y tf-2cos(a;+a) + e~14—2

2 1 2 THEORY OF HEAT. [CH. III. SECT. VII.

This equation contains, in real terms under a finite form, the

integral of the equation -7-3 + -7-3 = 0, applied to the problem of

the uniform movement of heat in a rectangular solid, exposed atits extremity to the constant action of a single source of heat.

It is easy to ascertain the relations of this integral to thegeneral integral, which has two arbitrary functions; these func-tions are by the very nature of the problem determinate, andnothing arbitrary remains but the function / ( a ) , consideredbetween the limits a = 0 and a = 7r. Equation (a) represents,under a simple form, suitable for numerical applications, the samevalue of v reduced to a convergent series.

If we wished to determine the quantity of heat which the solidcontains when it has arrived at its permanent state, we shouldtake the integral fdccfdy v from x = 0 to x = ir, and from y = 0 toy = oo; the result would be proportianal to the quantity required.In general there is no property of the uniform movement of heatin a rectangular plate, which is not exactly represented by thissolution.

We shall next regard problems of this kind from another pointof view, and determine the varied movement of heat in differentbodies.

CHAPTER IV.

OF THE LINEAR AND VARIED MOVEMENT OF HEAT IN A RING.

SECTION I.

General solution of the problem.

238. THE equation which expresses the movement of heatin a ring has been stated in Article 105; it is

dv K d2v hi n x

The problem is now to integrate this equation : we maywrite it simply

dv 7 d2v ,

dt dxl

wherein k represents ~yr, and h represents Ytnq> x denotes the

length of the arc included between a point m of the ring and theorigin 0, and v is the temperature which would be observed atthe point m after a given time t We first assume v = e~htu,

u being a new unknown, whence we deduce -rr — k -j—^; now this

equation belongs to the case in which the radiation is nul atthe surface, since it may be derived from the preceding equa-tion by making h = 0: we conclude from it that the differentpoints of the ring are cooled successively, by the action of themedium, without this circumstance disturbing in any manner thelaw of the distribution of the heat.

In fact on integrating the equation -T7 = ^ - T T J we shoulddt cCx

find the values of u which correspond to different points of the

214 THEORY OF HEAT. [CHAP. IV.

ring at the same instant, and we should ascertain what the stateof the solid would be if heat were propagated in it without anyloss at the surface; to determine then what would be the stateof the solid at the same instant if this loss had occurred, it willbe sufficient to multiply all the values of u taken at differentpoints, at the same instant, by the same fraction e~ht. Thus thecooling which is effected at the surface does not change the lawof the distribution of heat; the only result is that the tempera-ture of each point is less than it would have been without thiscircumstance, and the temperature diminishes from this causeaccording to the successive powers of the fraction e~ht.

239. The problem being reduced to the integration of the7 _72

equation -r- = k j - ^ , we shall, in the first place, select the sim-plest particular values which can be attributed to the variableu; from them we shall then compose a general value, and weshall prove that this value is as extensive as the integral, whichcontains an arbitrary function of x, or rather that it is thisintegral itself, arranged under the form which the problem re-quires, so that there cannot be any different solution.

It may be remarked first, that the equation is satisfied if wegive to u the particular value aemt sin nx, m and n being" subjectto the condition m = — kn*. Take then as a particular value ofu the function e~knH sin nx.

In order that this value may belong to the problem, it mustnot change when the distance x is increased by the quantity 27rr,r denoting the mean radius of the ring. Hence 2irnr must be a

multiple i of the circumference 2TT ; which gives n = - .

We may take i to be any integer; we suppose it to bealways positive, since, if it were negative, it would suffice tochaDge the sign of the coefficient a in the value ae-JcnHsmnx.

The particular value ae V<1 sin — could not satisfy the problem

proposed unless it represented the initial state of the solid. Now

on making t = 0, we find u = a sin — : suppose then that the

SECT. I.] PARTICULAR SOLUTIONS. 215

initial values of u are actually expressed by a sin - ; that is to

say, that the primitive temperatures at the different points areproportional to the sines of angles included between the radiiwhich pass through those points and that which passes throughthe origin, the movement of heat in the interior of the ring will

be exactly represented by the equation u = ae r<l sin - , and if

we take account of the loss of heat through the surface, we find

v = ae x r/ sin - .

In the case in question, which is the simplest of all those whichwe can imagine, the variable temperatures preserve their primi-tive ratios, and the temperature at any point diminishes accord-ing to the successive powers of a fraction which is the same forevery point.

The same properties would be noticed if we supposed theinitial temperatures to be proportional to the sines of the double

xof the arc - ; and in general the same happens when the given

temperatures are represented by a sin — , i being any integer

whatever.

We should arrive at the same results on taking for theparticular value of u the quantity ae"knHcos nx: here also we have

n

2nirr = 2tirf and n = - ; hence the equation

->* ixu = ae r cos —

r

expresses the movement of heat in the interior of the ring if the

initial temperatures are represented by cos — .

In all these cases, where the given temperatures are propor-

tional to the sines or to the cosines of a multiple of the arc - ,

the ratios established between these temperatures exist con-tinually during the infinite time of the cooling. The same would

216 THEOBY OF HEAT. [CHAP. IV.

be the case if the initial temperatures were represented by the

function a sin — + 6 cos —, i being any integer, a and b any co-T V

IX 7 IX

refficients whatever.

240. Let us pass now to the general case in which the initialtemperatures have not the relations which we have just supposed,but are represented by any function whatever Fix). Let us give

to this function the form cf> (-) , so that we have F(x) = <f>\*-), and

imagine the function $> ( - } to be decomposed into a series of

sines or cosines "of multiple arcs affected by suitable coefficients.We write down the equation

a, sin (o -) + at sin (1 - ) + a9 sin (2 - ) + &c.

! + b, cos 0 fj + \ cos (l fj + \ cos (2 ?J + &c.

The numbers a0, av a2..., 60, biy b2... are regarded as knownand calculated beforehand. It is evident that the value of u willthen be represented by the equation

n -

a. sin -i r

6 , COS —-

. sin z -

, cos z -r

e> 2

&C.

In fact, 1st, this value of u satisfies the equation -v- = k - —2,

since it is the sum of several particular values ; 2nd, it does notchange when we increase the distance x by any multiple whateverof the circumference of the ring;.3rd, it satisfies the initial state,since on making t = 0, we find the equation (e). Hence all theconditions of the problem are fulfilled, and it remains only tomultiply the value of u by e~ht-

241. As the time increases, each of the terms which composethe value of u becomes smaller and smaller; the system of tem-peratures tends therefore continually towards the regular and con-

SECT. I.] COMPLETE SOLUTION. 217

stant state in which the difference of the temperature u from theconstant b0 is represented by

a sin - + b cos - ) e r .r r/

Thus the particular values which we have previously considered,and from which we have composed the general value, derive theirorigin from the problem itself. Each of them represents anelementary state which could exist of itself as soon as it is sup-posed to be formed; these values have a natural and necessaryrelation with the physical properties of heat.

To determine the coefficients a0, al9 a2, &c, 60, bl9 \ , &c, wemust employ equation (II), Art. 234, which was proved in thelast section of the previous Chapter.

Let the whole abscissa denoted by X in this equation be 27rr,let x be the variable abscissa, and let fix) represent the initialstate of the ring, the integrals must be taken from x = 0 tox = 27rr; we have then

•f cos cos $

fx\ f foc\ f x\ f ( x\in (-j I sin l-jf(x)dx + sm (2 -J I sin I 2-jf(x) dx + &c.

Knowing in this manner the values of a0, aiy a2, &c,boJ ii9 h2, &c, if they be substituted in the equation we havethe following equation, which contains the complete solution ofthe problem :

7TVV = ( \x) dx

u

cos- (cos -fix) dx

sm2 - (sin— f{x)dxrj\ r J

cos 2 - cos — f(x)dx)r J \ r J

+ &c\.)

(E).


All the integrals must be taken from x = 0 to x = 2irr.

The first term ^— I fix) dx, which serves to form the value of

v, is evidently the mean initial temperature, that is to say, thatwhich each point would have if all the initial heat were distri-buted equally throughout.

242. The preceding equation (E) may be applied, whateverthe form of the given function fix) may be. We shall considertwo particular cases, namely: 1st, that which occurs when thering having been raised by the action of a source of heat to itspermanent temperatures, the source is suddenly suppressed; 2nd,the case in which half the ring, having been equally heatedthroughout, is suddenly joined to the other half, throughout whichthe initial temperature is 0.

We have seen previously that the permanent temperaturesof the ring are expressed by the equation v = ao? + boTx; the

-JE-value of quantity a being e ES, where I is the perimeter of thegenerating section, and S the area of that section.

If it be supposed that there is but a single source of heat, the(in

equation -7- = 0 must necessarily hold at the point opposite toax

that which is occupied by the source. The condition ao? — boTx= 0

will therefore be satisfied at this point. For convenience of calcu-

lation let us consider the fraction -TFTT to be equal to unity, and let

us take the radius r of the ring to be the radius of the trigono-metrical tables, we shall then have v = aex + be~x; hence the initialstate of the ring is represented by the equation

It remains only to apply the general equation (E), and de-noting by M the mean initial heat (Art. 241), we shall have

This equation expresses the variable state of a solid ring, whichhaving been heated at one of its points and raised to stationary

SECT. I.] FURTHER APPLICATION. 219

temperatures, cools in air after the suppression of the source ofheat.

243. In order to make a second application of the generalequation (E), we shall suppose,the initial heat to be so distributedthat half the ring included between x = 0 and x = ir has through-out the temperature 1, the other half having the temperature 0.I t is required to determine the state of the ring after the lapse ofa time t.

The function f(x)9 which represents the initial state, is in thiscase such that its value is 1 so long as the variable is includedbetween 0 and ir. I t follows from this that we must supposef(x) = l, and take the integrals only from x = 0 to x = ir, theother parts of the integrals being nothing by hypothesis. Weobtain first the following equation^ which gives the developmentof the function proposed, whosp value is 1 from x — 0 to X = TT andnothing from x = ir to x = 27r,

f(x) = ^ -i— (sinx + « sin 3x + - sin 5x + = sin 7x + &c.).It IT \ o 5 7 /

If now we substitute in the general equation the values whichwe have just found for the constant coefficients, we shall have theequation

-TTV = e~M (-r IT +'sin x e~kt + sin 3x e~w+7 sin 5x e~B2Jct + &c.V

which expresses the law according to which the temperature ateach point of the ring varies, and indicates its state after anygiven time: we shall limit ourselves to the two preceding applica-tions, and add only some observations on the general solutionexpressed by the equation (E).

244. 1st. If k is supposed infinite, the state of the ring is

expressed thus, 7rrv = e~ht^jf(x)dx> or, denoting by M the

mean initial temperature (Art. 241), v = e~uM. The temperatureat every point becomes suddenly equal to the mean temperature,and all the different points retain always equal temperatures,which is a necessary consequence of the hypothesis in which weadmit infinite conducibility.


2nd. We should have the same result if the radius of the ringwere infinitely small.

3rd. To find the mean temperature of the ring after a time t

we must take the integral \f{x)dx from x = 0 to x=2irr, and

divide by 2irr. Integrating between these limits the differentparts of the value of u, and then supposing x = 2irr, we find thetotal values of the integrals to be nothing except for the firstterm; the value of the mean temperature is therefore, after thetime ty the quantity e~htM. Thus the mean temperature of thering decreases in the same manner as if its conducibility were in-finite ; the variations occasioned by the propagation of heat in thesolid have no influence on the value of this temperature.

In the three cases which we have just considered, the tem-perature decreases in proportion to the powers of the fraction e~\or, which is the same thing, to the ordinate of a logarithmiccurve, the abscissa being equal to the time which has elapsed.This law has been known for a long time, but it must be remarkedthat it does not generally hold unless the bodies are of smalldimensions. The previous analysis tells us that if the diameter ofa ring is not very small, the cooling at a definite point would notbe at first subject to that law; the same would not be the casewith the mean temperature, which decreases always in proportionto the ordinates of a logarithmic curve. For the rest, it must notbe forgotten that the generating section of the ring is supposed tohave dimensions so small that different points of the same sectiondo not differ sensibly in temperature.

4th. If we wished to ascertain the quantity of heat whichescapes in a given time through the surface of a given portion of

the ring, the integral hi I dt I vdx must be employed, and must

be taken between limits relative to the time. For example,if we took 0 and 2ir to be the limits of x, and 0, oo , to be thelimits of t; that is to say, if we wished to determine the wholequantity of heat which escapes from the entire surface, during thecomplete course of the cooling, we ought to find after the integra-tions a result equal to the whole quantity of the initial heat, or2-TrrM, M being the mean initial temperature.

SECT. I.J DISTRIBUTION OF HEAT IN THE RING. 221

5th. If we wish to ascertain how much heat flows in a giventime, across a definite section of the ring, we must employ the

integral - KS I dt^-, writing for j - the value of that function,

taken at the point in question.

245. Heat tends to be distributed in the ring according toa law which ought to be noticed. The more the time whichhas elapsed increases the smaller do the terms which composethe value of v in equation (E) become with respect to thosewhich precede them. There is therefore a certain value of t forwhich the movement of heat begins to be represented sensiblyby the equation

s i n ~ + K c o s -r 1 r

The same relation continues to exist during the infinite timeof the cooling. In this state, if we choose two points of the ringsituated at the ends of the same diameter, and represent theirrespective distances from the origin by x1 and oc2, and their cor-responding temperatures at the time t by vx and v2; we shall have

/ x x \ - ?)ax sin — + bt cos - H e **> e ~h\

J + hx cos Jor T

The sines of the two arcs — and -f differ only in sign; the

same is the case with the quantities cos — and cos —; hence

thus the half-sum of the temperatures at opposite points givesa quantity aoe~ht, which would remain the same if we chose twopoints situated at the ends of another diameter. The quantityaoe~ht, as we have seen above, is the value of the mean tempera-ture after the time t Hence the half-sum of the temperatureat any two opposite points decreases continually with the meantemperature of the ring, and represents its value without sensibleerror, after the cooling has lasted for a certain time. Let us


examine more particularly in what the final state consists, whichis expressed by the equation

v = ]an+ a, sin - + b. cos - ) e

If first we seek the point of the ring at which we have thecondition

. oo 7 x ^ x , fb\a, sin - + k cos - = 0, or - = — arc tan - 1 1 ,1 r * r r v v

we see that the temperature at this point is at every instantthe mean temperature of the ring: the same is the case withthe point diametrically opposite; for the abscissa x of the latterpoint will also satisfy the above equation

X f U

~ — arc tan ( -r \ ax

Let us denote by X the distance at which the first of thesepoints is situated, and we shall have

. Xsin —

cos —r

and substituting this value of bv we have

{ ax . (x X\ -h±\ _u

cos — ir

If we now take as origin of abscissae the point which corre-sponds to the abscissa X, and if we denote by u the new abscissax — X, we shall have

/ u uv = e~ht (ao + b sin - e~ "

\ vAt the origin, where the abscissa u is 0, and at the opposite

point, the temperature v is always equal to the mean tempera-ture; these two points divide the circumference of the ring intotwo parts whose state is similar, but of opposite sign; each pointof one of these parts has a temperature which exceeds the meantemperature, and the amount of that excess is proportional tothe sine of the distance from the origin. Each point of the

SECT. 1.] PARTIAL CHANGES OF TEMPERATURE. 223

other part has a temperature less than the mean temperature,and the defect is the same as the excess at the opposite point.This symmetrical distribution of heat exists throughout the wholeduration of the cooling. At the two ends of the heated half,two flows of heat are established in direction towards the cooledhalf, and their effect is continually to bring each half of thering towards the mean temperature.

246. We may now remark that in the general equation whichgives the value of v, each of the terms is of the form

-vk4a, sin i - + b, cos i -r * r) e

We can therefore derive, with respect to each term, consequencesanalogous to the foregoing. In fact denoting by X the distancefor which the coefficient

a* sin 2 - + 6. cos i-• r r

X

is nothing, we have the equation b^ — ^ tan i — , and this sub-

stitution gives, as the value of the coefficient,(x-X\a sin ^ i\

a being a constant. It follows from this that taking the pointwhose abscissa is X as the origin of co-ordinates, and denotingby u the new abscissa x — X, we have, as the expression of thechanges of this part of the value of v, the function

If this particular part of the value of v existed alone, so as tomake the coefficients of all the other parts nul, the state of thering would be represented by the function

ae~ r2 sin [i - ,

and the temperature at each point would be proportional to thesine of the multiple i of the distance of this point from the origin.This state is analogous to that which we have already described:


it differs from it in that the number of points which have alwaysthe same temperature equal to the mean temperature of the ringis not 2 only, but in general equal to 2i. Each of these points ornodes separates two adjacent portions of the ring which are ina similar state, but opposite in sign. The circumference is thusfound to be divided into several equal parts whose state is alter-nately positive and negative. The flow of heat is the greatestpossible in the nodes, and is directed towards that portion whichis in the negative state, and it is nothing at the points which areequidistant from two consecutive nodes. The ratios which existthen between the temperatures are preserved during the whole ofthe cooling, and the temperatures vary together very rapidly inproportion to the successive powers of the fraction

If we give successively to i the values 0, 1, 2, 3, &c, we shallascertain all the regular and elementary states which heat canassume whilst it is propagated in a solid ring. When one of thesesimple modes is once established, it is maintained of itself, and theratios which exist between the temperatures do not change; butwhatever the primitive ratios may be, and in whatever mannerthe ring may have been heated, the movement of heat can be de-composed into several simple movements, similar to those whichwre have just described, and which are accomplished all togetherwithout disturbing each other. In each of these states the tempe-rature is proportional to the sine of a certain multiple of the dis-tance from a fixed point. The sum of all these partial temperatures,taken for a single point at the same instant, is the actual tempera-ture of that point. Now some of the parts which compose thissum decrease very much more rapidly than the others. It followsfrom this that the elementary states of the ring which correspondto different values of iy and whose superposition determines thetotal movement of heat, disappear in a manner one after theother. They cease soon to have any sensible influence on thevalue of the temperature, and leave only the first among them toexist, in which i is the least of all. In this manner we form anexact idea of the law according to which heat is distributed ina ring, and is dissipated at its surface. The state of the ring be-comes more and more symmetrical; it soon becomes confounded

SECT. II.] TRANSFER BETWEEN SEPARATE MASSES. 225

with that towards which it has a natural tendency, and which con-sists in this, that the temperatures of the different points becomeproportional to the sine of the same multiple of the arc whichmeasures the distance from the origin. The initial distributionmakes no change in these results.

SECTION II.

Of the communication of heat between separate masses.

247. We have now to direct attention to the conformity ofthe foregoing analysis with that which must be employed to de-termine the laws of propagation of heat between separate masses;we shall thus arrive at a second solution of the problem of themovement of heat in a ring. Comparison of the two results willindicate the true foundations of the method which we have fol-lowed, in integrating the equations of the propagation of heat incontinuous bodies. We shall examine, in the first place, an ex-tremely simple case, which is that of the communication of heatbetween two equal masses.

Suppose two cubical masses m and n of equal dimensions andof the same material to be unequally heated; let their respectivetemperatures be a and 5, and let them be of infinite conducibility.If we placed these two bodies in contact, the temperature in eachwould suddenly become equal to the mean temperature \ (a + b).Suppose the two masses to be separated by a very small interval,that an infinitely thin layer of the first is detached so as to bejoined to the second, and that it returns to the first immediatelyafter the contact. Continuing thus to be transferred alternately,and at equal infinitely small intervals, the interchanged layercauses the heat of the hotter body to pass gradually into thatwhich is less heated; the problem is to determine what would be,after a given time, the heat of each body, if they lost at their sur-face no part of the heat which they contained. We do not supposethe transfer of heat in solid continuous bodies to be effected in amanner similar to that which ,we have just described: we wishonly to determine by analysis the result of such an hypothesis.

Each of the two masses possessing infinite conducibility, thequantity of heat contained in an infinitely thin layer, is sud-

F. H. 15


denly added to that of the body with which it is in contact; and acommon temperature results which is equal to the quotient of thesum of the quantities of heat divided by the sum of the masses.Let co be the mass of the infinitely small layer which is separatedfrom the hotter body, whose temperature is a; let a and /3 be thevariable temperatures which correspond to the time t, and whoseinitial values are a and b. When the layer co is separated from themass m which becomes m — co9 it has like this mass the tempera-ture a, and as soon as it touches the second body affected with thetemperature /3, it assumes at the same time with that body a

temperature equal to —— —. The layer co, retaining the last

temperature, returns to the first body whose mass is m — co andtemperature a. We find then for the temperature after the secondcontact

, N /ra/3 + aoAa [m — co) + }co

v J \ m + co /am 4- pco

m

The variable temperatures a and /3 become, after the interval

dt, a. + (a — /3) —, and /3 + (a — /3) —; these values are found, by

suppressing the higher powers of co. We thus have

da = — (a — 6) — and dB = (a — 6) — :m K J m

the mass which had the initial temperature /3 has received in oneinstant a quantity of heat equal to md/3 or (a— /3) co, which hasbeen lost in the same time by the first mass. We see by thisthat the quantity of heat which passes in one instant from themost heated body into that which is less heated, is, all other thingsbeing equal, proportional to the actual difference of temperatureof the two bodies. The time being divided into equal intervals,the infinitely small quantity co may be replaced by kdt, k being thenumber of units of mass whose sum contains co as many times as

the unit of time contains dt, so that we have - = —. We thusco dt

obtain the equations

dx = -(a-/3)-dt and d/3 = (a -/3) - dt.

SECT. II.] RECIPKOCAL CONDUC1BILITY. 227

248. If we attributed a greater value to the volume co, whichserves, it may be said, to draw heat from one of the bodiesfor the purpose of carrying it to the other, the transfer wouldbe quicker; in order to express this condition it would benecessary to increase in the same ratio the quantity k whichenters into the equations. We might also retain the valueof co and suppose the layer to accomplish in a given time agreater number of oscillations, which again would be indicatedby a greater value of k. Hence this coefficient represents in somerespects the velocity of transmission, or the facility with whichheat passes from one of the bodies into the other, that is to say,their reciprocal conducibility.

249. Adding the two preceding equations, we have

dx + d/3 = 0,

and if we subtract one of the equations from the other, we have

dx - d,8 -1-2 (a — /3) — dt= 0, and, making a — /3 = y,?7Z

dy + 2-ydt = 0.

Integrating and determining the constant by the condition that

the initial value is a - 6, we have y = (a — b) e m . The differ-ence y of the temperatures diminishes as the ordinate of a loga-

JLk

rithmic curve, or as the successive powers of the fraction e m .As the values of a and /3, we have

1 1 _?** 1 1 ? ^« = g (a + h) - 2 (a ~ J ) e~ m ' ^ = 2 ( a + ^ + 2 ^ ~ b^e' m '

250. In the preceding case, we suppose the infinitely smallmass «, by means of which the transfer is effected, to be alwaysthe same part of the unit of mass, or, which is the same thing,we suppose the coefficient k which measures the reciprocal con-ducibility to be a constant quantity. To render the investigationin question more general, the constant k must be consideredas a function of the two actual temperatures a and /?. We should

then have the two equations dx = — (OL —j3)—dt, and

15—2

in which k would be equal to a function of a and 0, which wedenote by <f> (a, 0). I t is easy to ascertain the law whichthe variable temperatures a and 0 follow, when they approachextremely near to their final state. Let y be a new unknownequal to the difference between a and the final value which is

- (a +1) or c. Let z be a second unknown equal to the difference

c-0. We substitute in place of a and 0 their values c-y andc — z; and, as the problem is to find the values of y and zy

when we suppose them very small, we need retain in the resultsof the substitutions only the first power of y and z. We thereforefind the two equations,

-dy = -(z-y)-tf>(c-y, c-z)dt

kand - dz = — (z — y) j> (c — y, c — z) dt,

developing the quantities which are under the sign (j> and omit-

ting the higher powers of y and z. We find dy=(z — y) — <f>dt,

kand dz = — [z — y) — cf)dt The quantity <f> being constant, itfollows that the preceding equations give for the value of thedifference z — y,a, result similar to that which we found above forthe value of a — 0.

From this we conclude that if the coefficient Jc, which wasat first supposed constant, were represented by any functionwhatever of the variable temperatures, the final changes whichthese temperatures would experience, during an infinite time,would still be subject to the same law as if the reciprocal con-ducibility were constant. The problem is actually to determinethe laws of the propagation of heat in an indefinite number ofequal masses whose actual temperatures are different.

251. Prismatic masses n in number, each of which is equalto m, are supposed to be arranged in the same straight line,and affected with different temperatures a, b, c, d, &c.; infinitely

SECT. II.] EQUAL PK1SMATIC MASSES IN LINE. 229

thin layers, each of which has a mass co, are supposed to beseparated from the different bodies except the last, and areconveyed in the same time from the first to the second, fromthe second to the third, from the third to the fourth, and soon; immediately after contact, these layers return to the massesfrom which they were separated; the double movement takingplace as many times as there are infinitely small instants dt; itis required to find the law to which the changes of temperatureare subject.

Let a, /3, 7, S,... a>, be the variable values which correspond tothe same time t, and which have succeeded to the initial valuesa, 6, c, d, &c. When the layers co have been separated from then — 1 first masses, and put in contact with the neighbouringmasses, it is easy to see that the temperatures become

a(m — co) /3"(m —ft))-}-ao> <y (m — co)-\-ftcom — co ' m ' m

S (m — to) + 7ft) met)

or,

a, £ + ( a - / 3 ) - , 7 + ( / 9 - 7 ) ^ , 8 + ( 7-8)- , . . .eo + (^-ft>)-.

When the layers co have returned to their former places,we find new temperatures according to the same rule, whichconsists in dividing the sum of the quantities of heat by the sumof the masses, and we have as the values of a, /3, 7 , S, &c, afterthe instant dt,

CO , . . C O7 + ( / 3 - Y - Y - £ ) - > • • • m

The coefficient of — is the difference of two consecutive dif-7)1

ferences taken in the succession a, /3, 7 , . . . ^ co. As to the first

and last coefficients of —, they may be considered also as dif-

ferences of the second order. It is sufficient to suppose the term

a to be preceded by a term equal to a, and the term co to be


followed by a term equal to ft). We have then, as formerly, onsubstituting kdt for ft), the following equations :

dco — —dt {(ft) — ft)) — (ft) —

252. To integrate these equations, we assume, according tothe known method,

a = a/\ /3 =•• a/\ 7 = a/\ . . . ft) = ane*';

h1,a1, a2, a3>... aw, being constant quantities which must be deter-mined. The substitutions being made, we have the followingequations:

k

k

A;

If we regard ax as a known quantity, we find the expressionfor a2 in terms of ax and h, then that of a3 in a2 and A-; the sameis the case with all the other unknowns, a4, a6, &c. The first andlast equations may be written under the form

a n d

m

k

SECT. II.] FORM OF THE SOLUTION. 231

Retaining the two conditions a = at and an = an+1, the valueof a2 contains the first power of h, the value of a3 contains thesecond power of h, and so on up to a>n+l, which contains thento power of h. This arranged, an+1 becoming equal to an, wehave, to determine h, an equation of the nth degree,, and at re-mains undetermined.

It follows from this that we shall find n values for A, and inaccordance with the nature of linear equations, the general valueof OL is composed of n terms, so that the quantities a, /3, 7, ... &c.are determined by means of equations such as

a = a/' + a;em + a??* + &c,

co = aneht + ay* + a^f* + &c.

The values of h} h!, h"> &c. are n in number, and are equal tothe n roots of the algebraical equation of the nth degree in h,which has, as we shall see further on, all its roots real.

The coefficients of the first equation av a/, a"> a"\ &c, arearbitrary; as for the coefficients of the lower lines, they are deter-mined by a number n of systems of equations similar to the pre-ceding equations. The problem is now to form these equations.

253. Writing the letter q instead of - r - , we have the fol-

lowing equations

We see that these quantities belong to a recurrent serieswhose scale of relation consists of two terms (q + 2) and - 1. We


can therefore express the general term am by the equation

am — A sin mu + B sin (m — 1) u,

determining suitably the quantities A, B, and u. First we findA and B by supposing m equal to 0 and then equal to 1, whichgives ao = B sin u} and ax — A sin u, and consequently

a = at sin mu r-1- sin (m — 1) u.

Substituting then the values of

«„> am-l> am-2> &C«

in the general equation

we find

sin mu = (q f 2) sin (w — 1) w — sin (m — 2) ^,

comparing which equation with the next,

sin mu = 2 cos u sin (m — 1) u — sin (m — 2) ^,

which expresses a known property of the sines of arcs increasingin arithmetic progression, we conclude that q + 2 = cos u, orq = — 2 versin w; it remains only to determine the value of thearc u.

The general value of am being

-r-1- [sin M — sin (m -1) u],

we must have, in order to satisfy the condition an+1= an, theequation

sin (n + 1) w — sin u = sin ?m - sin (w- — 1) u,

whence we deduce sin nu = 0, or u = i - , 7r being the semi-

circumference and i any integer, such as 0, 1, 2, 3, 4, ... (% —1);

thence we deduce the n values of q or - ^ . Thus all the roots

of the equation in h, which give the values of h, li, Ji\ h"\ &c.are real and negative, and are furnished by the equations

SECT. II.] PARTICULAR TEMPERATURE-VALUES. 233

k= _ 2 — versin (0 - ) ,

in \ nj

K = — 2 — versinm

hh" = - 2 -versing - V

7/7.-11 rx k • f/ is 7r)

rc } = — 2 — v e r s m ^ (w — 1) - } .

Suppose then that we have divided the semi-circumference irinto n equal parts, and that in order to form u, we take i of thoseparts, i being less than n, we shall satisfy the differential equationsby taking ax to be any quantity whatever, and making

SIIIM ~ Sin OM - ^ versin u

^ e m

1 sm wSin 2l^ — Sin 1 ^ -^versinu

2 sini* '

s i n Su — s i n 2 ^ - — versin ?*7 = a . m1

smw

s m n u — s m ( n — l ) u v1 sin u e

As there are n different arcs which we may take for u3

namely,

there are also n systems of particular values for a, /3, 7, &c,and the general values of these variables are the sums of theparticular values.

254 "We see first that if the arc u is nothing, the quantitieswhich multiply a, in the values of a, /3, 7, &cv become all equal

sin u — sin Ou , , ... -. -, i ±1

to unity, since : takes the value 1 wnen the arc uJ sm u

vanishes; and the same is the case .with the quantities which are

found in the following equations. From this we conclude thatconstant terms must enter ioto the general values of a, /?, 7, ... <y.

Further, adding all the particular values corresponding toa, j3, 7, ... &a, we have

sin nu -—

an equation whose second member is reduced to 0 provided thearc u does not vanish; but in that case we should find n to be

the value of — . We have then in generalsin u to

now the initial values of the variables being a, b, c, &c, we mustnecessarily have

naJ = a + h + c + &c.;

it follows that the constant term which must enter into each ofthe general values of

a, A % ••• « is - (a + b + c +&c),

that is to say, the mean of all the initial temperatures.

As to the general values of a, ft 7, ... co, they are expressedby the following equations :

a = ~(a + b + c+ &c.) + Ojj esm u

Sin U — Sin Oil' - — versin u': ; e m

sm u

Sin V," — Sin Ou+ i T , esm u+ &c,

1 , , 7 . , n \ Sill 2U— Sin U -—yevshiu- ( a + 6 + c + &c.) + a, ^ e m

n 7 sin u

7 Sin 2 U — S in i ^ - — versin tt'+ &! :—7 e m

sin 2u" — sin u" - -^ venm tt"+ Cj : 77 6 m

sm u

SECT. II.] GENERAL TEMPERATURE-VALUES. 235

1 , , 7 , , p N sin Su - s i nry = - ( a + 6 + c + &c.) +' nK J

e.

n

sin 3u' — sin 2u' - ^ v, : ; e

sin wsin Su" - sin

1 sin u+ &c,

o N / S i n 7116 — Sin (n — 1) tt\ - ^ v e&c.) + a, =— — e m

\ sin u J

, /sin nu — sin (w — 1) w'\ ~ v+ h[ =—/ )e

1 \ smu J/Vsin nu" — sin (n — T) u"\ - — v

=—\, }-— e m

sm u J

+ &c.

255. To determine the constants a, b, c, d...&c, we mustconsider the initial state of the system. In fact, when the timeis nothing, the values of a} ft, y, &c. must be equal to a, 6, c, &c;we have then n similar equations to determine the n constants.Tie quantities *

sinw—sin(h£, sin2&—sinw, sin3& — sin2^, . . . , sin nu—sin (n— 1) u,

may be indicated in this manner,

AsinO^, Asinw, A sin 2u, A sin3w,... A s'm(n— 1) u;

the equations proper for the determination of the constants are,if the initial mean temperature be represented by C,

A sin u 7 A sin u' A sin u" 0

— +bl——T +cx —. — + & c ,sm u sin u sm u

A sin 2u , A sin 2u A sin 2u" p

—. + K —.-—— + o —= 77- + &c,sm u sm u sm u

n t A sin Su , A sin Su , A sin Su"G+ aY : + b . 7— + G : 77- + &C,

sm u sm u sm u


The quantities a,, b19 cv d19 and G being determined by theseequations, we know completely the values of the variables

a, ft 7, 8, ...ty.

We can in general effect the elimination of the unknowns inthese equations, and determine the values of the quantitiesa, b, c, d, &a, even when the number of equations is infinite; weshall employ this process of elimination in the following articles.

256. On examining the equations which give the generalvalues of the variables a, ft 7 co, we see that as the timeincreases the successive terms in the value of each variable de-crease very unequally: for the values of u, u, u"9 u", &c. being

I ? , 2 ? , 3 ? , 4 - , &c.,n n n n'

the exponents versin u, versin u, versin u"9 versin u", &c.become greater and greater. If we suppose the time t to beinfinite, the first term of each value alone exists, and the tempera-ture of each of the masses becomes equal to the mean tempera-ture - (a + b.+ c+...&c). Since the time t continually increases,

each of the terms of the value of one of the variables diminishesproportionally to the successive powers of ,a fraction which, for the

versin u . versin M'

second term, is e m , for the third term e m , and so on.The greatest of these fractions being that which corresponds tothe least of the values of u, it follows that to ascertain the lawwhich the ultimate changes of temperature follow, we need con-sider only the two first terms; all the others becoming incom-parably smaller according as the time t increases. The ultimatevariations of the temperatures a, ft, 7, &c. are therefore expressedby the following equations :

1 / , 7 , . 0 \ Sin W — Sin Olt ~ versin M

a = - ( a + J + c + &cO + a1 ^ e -

7 = - {a + o + c + &c.){ + + c + ) + x 7

SECT. II.] CONCLUDING TEMPERATURES. 237

257. If we divide the semi-circumference into n equal parts,and, having drawn the sines, take the difference between twoconsecutive sines, the n differences are proportional to the eo-

_ ?5? versin u

efficients of e m , or to the second terms of the values ofa, ft, 7,...ct). For this reason the later values of a, A Y-«G> aresuch that the differences between the final temperatures and the

mean initial temperature - (a + b + c + &c.) are always propor-tional to the differences of consecutive sines. In whatevermanner the masses have first been heated, the distribution ofheat is effected finally according to a constant law; If wemeasured the temperatures in the last stage, when they differlittle from the mean temperature, we should observe that thedifference between the temperature of any mass whatever and themean temperature decreases continually according to the succes-sive powers of the same fraction; and comparing amongst them-selves the temperatures of the different masses taken at the sameinstant, we should see that the differences between the actualtemperatures and the mean temperature are proportional to thedifferences of consecutive sines, the semi-circumference havingbeen divided into n equal parts.

258. If we suppose the masses which communicate heat to eachother to be infinite in number, we find for the arc u an infinitelysmall value; hence the differences of consecutive sines, taken onthe circle, are proportional to the cosines of the corresponding

- sin mu — sin (m — \)u. , , ,arcs- for : — is equal to cos mu, when t he

9 sin uarc u is infinitely small. In this case, the quantities whose tem-peratures taken at the same instant differ from the mean tempera-ture to which they all must tend, are proportional to the cosineswhich correspond to different points of the circumference dividedinto an infinite number of equal parts. If the masses whichtransmit heat are situated at equal distances from each other onthe perimeter of the semi-circumference 7r, the cosine of the arc atthe end of which any one mass is placed is the measure of thequantity by which the temperature of that mass differs yet fromthe mean temperature. Thus the body placed in the middle ofall the others is that which arrives most quickly at. that mean


temperature; those which are situated on one side of the middle,all have an excessive temperature, which surpasses the meantemperature the more, according as they are more distant fromthe middle; the bodies which are placed on the other side, allhave a temperature lower than the mean temperature, and theydiffer from it as much as those on the opposite side, but in con-trary sense. Lastlj7, these differences, whether positive or negative,all decrease at the same time, proportionally to the successivepowers of the same fraction; so that they do not cease to be repre-sented at the same instant by the values of the cosines of thesame semi-circumference. Such in general, singular cases ex-cepted, is the law to which the ultimate temperatures are subject.The initial state of the system does not change these results. Weproceed now to deal with a third problem of the same kind as thepreceding, the solution of which will furnish us with many usefulremarks.

259. Suppose n equal prismatic masses to be placed at equaldistances on the circumference of a circle. All these bodies,enjoying perfect conducibility, have known actual temperatures,different for each of them; they do not permit any part of theheat which they contain to escape at thpir surface; an infinitelythin layer is separated from the first mass to be united to thesecond, which is situated towards the right; at the same time aparallel layer is separated from the second mass, carried from leftto right, and joined to the third; the same is the case with all theother masses, from each of which an infinitely thin layer is sepa-rated at the same instant, and joined to the following mass.Lastly, the same layers return immediately afterwards, and areunited to the bodies from which they had been detached.

Heat is supposed to be propagated between the masses bymeans of these alternate movements, which are accomplishedtwice during each instant of equal duration; the problem is tofind according to what law the temperatures vary : that is to say,the initial values of the temperatures being given, it is required toascertain after any given time the new temperature of each of themasses.

We shall denote by av a2, a^...av..an the initial temperatureswhose values are arbitrary, and by av ct2, or3...a4...orn the values of

SECT. II.] EQUAL PRISMATIC MASSES IN CIRCLE. 239

the same temperatures after the time t has elapsed. Each of thequantities a is evidently a function of the time t and of all theinitial values av a2, as...an: it is required to determine thefunctions a.

260. We shall represent the infinitely small mass of the layerwhich is carried from one body to the other by co. We mayremark, in the first place, that when the layers have been separatedfrom the masses of which they have formed part, and placed re-spectively in contact with the masses situated towards the right,the quantities of heat contained in the different bodies become(m - co) a1 + coan, (m - to) a2 + coav (m - co) a3 + wa2,..., (m - co) an

+ <°xn-i y dividing each of these quantities of heat by the mass m,we have for the new values of the temperatures

co / v , co , x co . x

a>+S^•-i"**)'•"and K + S n-x"a^;

that is to say, to find the new state of the temperature after thefirst contact, we must add to the value which it had formerly the

product of — by the excess of the temperature of the body

from which the layer has been separated over that of the body towhich it has been joined. By the same rule it is found that thetemperatures, after the second contact, are

CO , N CO . x

The time being divided into equal instants, denote by dt theduration of the instant, and suppose co to be contained in kunits of mass as many times as dt is contained in the units oftime, we thus have co = kdt. Calling dzv d<x2, d^...dcLif^.dau the

infinitely small increments which the temperatures av a2,...a....crtt

receive during the instant dt, we have the following differentialequations :

klit

kda2 = — dt ( 2 + )a2

772,k

kn_x = - dt (<xn_2 -

261. To solve these equations, we suppose in the first place,according to the known method,

The quantities bv &2, 63, ... bn are undetermined constants, asalso is the exponent h. It is easy to see that the values ofa1? c/2,... an satisfy the differential equations if they are subject tothe following conditions:

Let q = -j—, we have, beginning at the last equation,

2 ) - & , ,+ 2 ) - * M ,

SECT. II.] PARTICULAR SOLUTION. 241

It follows from this that we may take, instead of bvb2,b3J...J.,...6w,the n consecutive sines which are obtained by dividing thewhole circumference 2ir into n equal parts. In fact, denoting the

arc 2 - by % the quantities

sin Ou, sin lu, sin 2u, sin 3w,... , sin (n - 1) u,

whose number is n} belong, as it is said, to a recurring serieswhose scale of relation has two terms, 2 cos u and — 1 : so thatwe always have the condition

sin in = 2 cos u sin (i —l)u — sin (i — 2) u.

Take then, instead of b19 62, ft8>... Jn, the quantities

sin Ow, sin lw, sin 2u, ... sin (w — 1) u,

and we have

q -f 2 = 2 cos w, g' = — 2 versin w, or q = — 2 versin —.

We have previously written q instead of -r-, so that the value

2k 2TTof A is versin —; substituting in the equations these values

m n ° l

of b. and h we have_?*£ versin ^

a, = smOue m

a2 = sin lue m

aQ = sin 2ue

= sin (w —

262. The last equations furnish only a very particular solu-tion of the problem proposed ; for if we suppose t = 0 we have, asthe initial values of a19 a2, a3,... an, the quantities

sin 0uy sin 1^, sin 2% ... sin (n— 1) u,

which in general differ from the given values av a2, a8,...an:but the foregoing solution deserves to be noticed because it ex-presses, as we shall see presently, a circumstance which belongs toall possible cases, and represents the ultimate variations of the

F. H. 16


temperatures. We see by this solution that, if the initial tern-peratures av a2, as1 ... an, were proportional to the sines

s i n 0 , rini, S i n 2 , ()^,n n n n

they would remain continually proportional to the same sines, andwe should have the equations

ao =

= « , / • " ,

. 2k . 2TTwhere h = — versin — .

For this reason, if the masses which are situated at equal dis-tances on the circumference of a circle had initial temperaturesproportional to the perpendiculars let fall on the diameterwhich passes through the first point, the temperatures wouldvary with the time, but remain always proportional to those per-pendiculars, and the temperatures would diminish simultaneouslyas the terms of a geometrical progression whose ratio is the

—— veraln —

fraction e m n .

263. To form the general solution, we may remark in thefirst place that we could take, instead of &1? b2, b3, ... bn, the ncosines corresponding to the points of division of the circumferencedivided into n equal parts. The quantities cos Ou, cos lu} cos 2u>...

cos (n — 1) u, in which u denotes the arc — , form also a recurring

series whose scale of relation consists of two terms, 2 cos u and — 1,for which reason we could satisfy the differential equations bymeans of the following equations,

Wet versin u

at = cos Oue m ,n versin u

a2 = cos lue m ,

a3 = c o s 2 u e m ver nM?

a n = cos ( n — l ) u e

SECT. II.] OTHER SOLUTIONS. 243

Independently of the two preceding solutions we could selectfor the values of bl9 b2, bs,... bn, the quantities

sinO.2w, sinl.2i£, sin2.2w, sin3.2w, ..., sin(w — l)2u\

or elsei~*f\Q I I /'it f*C\Q • sit £~*f\Q$ / y f l / C*f\C* -£ V 4 / /"*/*fcC! ((V\ I 1 /tt

In fact, each of these series is recurrent and composed of nterms; in the scale of relation are two terms, 2 cos 2u and — 1;and if we continued the series beyond n terms, we should find nothers respectively equal to the n preceding.

In general, if we denote the arcs

02TT n 2TT O 2ir . N 2ir Q

—, 1—, 2—--,..., {n-1)—, &c,n n n x ; n

by ux, u2, u3,..., un, we can take for the values of bl9 b2, 63, ... bn

the n quantities,

sinOi/^, sin lu.f sin 2uif sin 3^f, ..., s i n ( n - l ) ^ ;

or else

cos 0^, cos lui9 cos 2ui9 cos 3nv ..., cos (n — 1) u{.

The value of h corresponding to each of these series is given by theequation

2kh = versin uf.m

We can give n different values to i, from i = 1 to i = n.

Substituting these values of bl9 b2> b3 ... bn9 in the equationsof Art. 261, we have the differential equations of Art. 260 satisfiedby the following results:

_?*? Versin u{ ——^versin ut

ax — sin Ow.e m , or at = cos Ou.e m ,. — — versin u$ versin U{

2 — S i n ± O^ c? , Kg — t U o ±11^ ,

— — versin w$ _ versin Wf

a3 = sin 2ufi , a3 = cos 2w.e w ,

- ^ * versin iH , - ^ versin ta = sin m — 1)u.e m , an = cos (w — l)u,e

16—2

264. The equations of Art. 260 could equally be satisfied byconstructing the values of each one of the variables <xl9 a2, a3, ... an

out of the sum of the several particular values which have beenfound for that variable; and each one of the terms which enterinto the general value of one of the variables may also be mul-tiplied by any constant coefficient. It follows from this that,denoting by Av Bv A2, i?2, A3, B3, ...*An9 Bn, any coefficientswhatever, we may take to express the general value of one of thevariables, am+1 for example, the equation

2M—— versin u^

am+1 = (Ax sin mut + Bx cos mu2) e

versin w2

+ [A1 sin mu2 + B2 cos mu2) e m

+ (An sin mun + Bn cos mun) e

The quantities Av A2J A3,... An> Bv B2> B3>... Bn, whichenter into this equation, are arbitrary, and the arcs u19 u2,u3, ... un

are given by the equations:

A 2?r ,277 2TT . - . 2TT

ut = 0 — , u2 = 1 — , u3 = 2 — , ..., un=(n-~l)—.

The general values of the variables ax, a2, a8, ... an are thenexpressed by the following equations:

- ( A

+ (^,

+ (A+ &C.

= (A

+ (A

+ (A+ &C.

sin O j

sin 0&2

sin 0w3

sin \ux

sin lw2

sin \u3

>

+ Bl

+ •».

COS

COS

COS

COS

COS

COS

OwJ e

0a2) e"

0w3) e

l iO e

! » > •

lu,)e

2&f versin %

/versin w2

.!** versin e/3

—^r versin ««i

versin «2

- - - versin w3

SECT. II.] GENERAL SOLUTION. 245

a3 = (Ax sin 2ux -f B1 cos 2ut) e

+ (A2 sin 2u2 + B2 cos 2uc

+ (AQ sin 2u» + B» cos 2u<_ ? M versin ?/3

+ &c.;

c?w = [Ax sin (w — 1) nx + 5X cos (n — 1) u^ e

+ [A2 sin (w — 1) u2 + ,B2 cos (n - 1) u2) e m

+ {A3 sin (?i — 1) u3 + ^ 3 cos (/? — 1) w8} e

265. If we suppose the time nothing, the values ava2,or3,... aw

must become the same as the initial values a17a2,a3, ... an. Wederive from this n equations, which serve to determine the coeffi-cients Av Bv A2, B2, As, B3, ... . It will readily be perceived thatthe number of unknowns is always equal to the number of equa-tions. In fact, the number of terms which enter into the valueof one of these variables depends on the number of differentquantities versing, versing, versing, &c, which we find ondividing the circumference 2ir into n equal parts. Now the

, - . . . . . A2TT . 1 2TT . 2TT Onumber 01 quantities versm 0—, versm 1 — , versm 2 — , &c.

^ n n nis very much less than n, if we count only those that aredifferent. Denoting the number n by 2i + 1 if it is odd,and by 2i if it is even, i + 1 always denotes the numberof different versed sines. On the other hand, when in the

„ .... . „ 27r . ^ 2TT . ft 2TT nseries 01 quantities versin 0 — , versm 1 — , versm 2 — , &c,

^ n n n 9

we come to a versed sine, versin X— , equal to one of the former

o

versin V — , the two terms of the equations which contain this

versed sine form only one term; the two different arcs uk andu\n which have the same versed sine, have also the same cosine,and the sines differ only in sign. It is easy to see that thearcs ux and u^, which have the same versed sine, are such that

the cosine of any multiple whatever of uK is equal to the cosineof the same multiple of u^, and that the sine of any multipleof uk differs only in sign from the sine of the same multipleof %'. It follows from this that when we unite into one thetwo corresponding terms of each of the equations, the two un-knowns AK and A^, which enter into these equations, are replacedby a single unknown, namely A^ — A^. As to the two unknownBk and Bx they also are replaced by a single one, namely BK + By :it follows from this that the number of unknowns is equal in allcases to the number of equations; for the number of terms isalways i + 1. We must add that the unknown A disappears ofitself from the first terms, since it is multiplied by the sine ofa nul arc. Further, when the number n is even, there is foundat the end of each equation a term in which one of the unknownsdisappears of itself, since it multiplies a nul sine; thus thenumber of unknowns which enter into the equations is equalto 2 ( i + l ) — 2, when the number n is even; consequently thenumber of unknowns is the same in all these cases as the numberof equations.

266. To express the general values of the temperaturesalf a2, <73 ... an, the foregoing analysis furnishes us with the equa-tions

2.ra = (A.WIO.O — -f-^cosO.O —1 V n x n

. l ^ c o s O . l ^ -?

n * n

4- (-4,sin0.2 hiLcos0.2 — )e

aa = (A. sin 1. 0 — + B, cos 1 . 0 — ) e\ n x n

. / i . ^ o 2?T n 2TT\+ Ld3 sin 2 .2 —• + JB. cos 2 .2 — ]

V n 3 n J+ &c,

n J

\ - ^ versin 2 ?5m

SECT. II.] FORM OF THE GENERAL SOLUTION. 247

* = ( A s i n 2 . 0 — + 5 l c o s 2 . 0 — ) e •

2 sin 2 . 1 — + B2 cos 2 . 1 ^ J ^""

+ (^1, sin 2. 2 — + 53 cos 2 .2 — ] e "

+ &c,

a = I ^ 1 s i n ( n - l ) 0 —+ 5 1cos(n-l)0—I.

sin (n — 1) 1 — 4- B2 cos (n —1)1 — [^

sin ( » - 1 ) 2 — + 5 , 0 0 8 ( ^ - 1 ) 2 —}-e TO u

To form these equations, we must continue in each equation

the succession of terms which contain versin 0 — , versin 1 — ,n n

versin 2 — , &c. until we have included every different versed

sine; and we must omit all the subsequent terms, commencingwith that in which a versed sine appears equal to one of thepreceding.

The number of these equations is n. If n is an even numberequal to 2i, the number of terms of each equation is i + 1; if nthe number of equations is an odd number represented by 2i+ 1,the number of terms is still equal to i +1. Lastly, among thequantities Av Blf A2, B2, &c, which enter into these equations,there are some which must be omitted because they disappear ofthemselves, being multiplied by nul sines.

267. To determine the quantities Av Bv A2, B2, A3, Bs, &c,which enter into the preceding equations, we must consider theinitial state which is known: suppose t = 0, and instead ofav a2> a8, &c, write the given quantities av a2, a3, &c, which arethe initial values of the temperatures. We have then to determineAv Bv A2> i?2, As9 Bs) &c, the following equations:


ax = Ax sin 0 . 0 — + A9 sin 0 .1 — + A3 sin 0. 2 — + &c.1 1 n 2 n n

9TT 9'7T 27T

+ 5 , cos 0. 0 — + B, cos 0 . 1 — + B. cos 0 . 2 — + &c.1 n 2 n s n

a2 = ^ sin 1.0 — + A, sin 1.1 — + A, sin 1.2 — + &c.

+ 5lCos 1.0 — + 5 , cos 1.1 — + 5 , cos 1.2 — + &c.1 w 2 n s n

a3 = .4 sin 2 .0 — + A, sin 2 .1 — + ^1, sin 2 .2 — + &c.1 ?i 2 » • 3 n

B cos 2. 0 \- B9 cos1 n 2

2-7T

B1cos(n-l)0-—+B2c

2 2TT ^ c c

rin(n-l)l —+

>s 2.2 — + &c.

J 3 s i n ( ^ - l ) 2 ^ + &c.

5 8 cos(n- l )2 2 ^ + &c.

.... (ml

268. In these equations, whose number is w, the unknownquantities are A19 B19 A29 B2, A3, J?3, &c., and it is required toeffect the eliminations and to find the values of these unknowns.We may remark, first, that the same unknown has a differentmultiplier in each equation, and that the succession of multiplierscomposes a recurring series. In fact this succession is that of thesines of arcs increasing in arithmetic progression, or of the cosmesof the same arcs; it may be represented by

sin Ou, sin luy sin 2u, sin SuP ... sin (n — 1) u,

or by cos Ou, cos Iw, cos 2u, cos Su, ... cos (n — 1) u.

The arc u is equal to i(—J if the unknown in question is Ai+1

or Bi+1. This arranged, to determine the unknown Ai+1 by meansof the preceding equations, we must combine the succession ofequations with the series of multipliers, sin Ou, sin lu, sin 2u,sin Su, ... sin(n — l)u, and multiply each equation by the cor-responding term of the series. If we take the sum of the equa-

SECT. II.] DETERMINATION OF COEFFICIENTS. 249

tions thus multiplied, we eliminate all the unknowns, exceptthat which is required to be determined. The same is the caseif we wish to find the value of Bi+1; we must multiply eachequation by the multiplier of Bi+1 in that equation, and then takethe sum of all the equations. It is requisite to prove that byoperating in this manner we do in fact make all the unknownsdisappear except one only. For this purpose it is sufficient to shew,firstly, that if we multiply term by term the two following series

sin Ou, sin lu, sin 2u, sin Su, ... sin (n — 1) u,

sin Ov, sin lv, sin 2v, sin 3v, ... sin (n — l)v,

the sum of the products

sin Ou sin Ov + sin lu sin lv + sin 2u sin 2v -f &c.

is nothing, except when the arcs u and v are the same, eachof these arcs being otherwise supposed to be a multiple of a part

of the circumference equal to — \ secondly, that if we multiply

term by term the two series

cos Ou, cos lu, cos 2u, ... cos (n — 1) u,

cos Ov, cos lv, cos 2v, ... cos (n — 1) v,

the sum of the products is nothing, except in the case whenu is equal to v \ thirdly, that if we multiply term by term the twoseries


cos Ov, cos lv, cos 2v, cos Sv, ... cos (n — 1) v>

the sum of the products is always nothing.

269. Let us denote by q the arc — , by jnq the arc u, and by

vq the arc v; /JL and v being positive integers less than n. Theproduct of two terms corresponding to the two first series willbe represented by

. . . . 1 . , N 1 . ,smjvq, or ( ) { i )

the letter ; denoting any term whatever of the series 0, 1, 2, 3,.,


(n - 1); now it is easy to prove that if we give to j its n successivevalues, from 0 to [n — 1), the sum

^cosO(>--z/)g + 2 C0S 1 ^ ~ 0 2 + ^ C O S 2 ( A 6 ' ~ ^ ) 2

+ | cos 3 (/*- v) q + ... + gcos ( n - 1) (/*-i>) j

has a nul value, and that the same is the case with the series

22 cos 0 0 + v) q + g cos 1 (fi +1/) j + ^ c o s 2

+ 2 cos 3

In fact, representing the arc {^-v)q by or, which is consequently

a multiple of — , we have the recurring series

cos 0a, cos lay cos 2a,... cos (w — 1) a,

whose sum is nothing.

To shew this, we represent the sum by s, and the two terms ofthe scale of relation being 2 cos a and — 1, we multiply successivelythe two members of the equation

s = cos 0a + cos 2a + cos 3a + ... + cos (n — 1) a

by — 2 cos a and by 4-1 ; then on adding the three equations wefind that the intermediate terms cancel after the manner of re-curring series.

If we now remark that not. being a multiple of the whole cir-cumference, the quantities cos (n — l)a, cos (n — 2) a, cos (n — 3) a,&c. are respectively the same as those which have been denotedby cos (— a), cos (— 2a), cos ( - 3a), ... &c. we conclude that

2s - 25 cos a = 0 ;

thus the sum sought must in general be nothing. In "the sameway we find that the sum of the terms due to the development ofh cos j {p + v)q is nothing. The case in which the arc representedby a is 0 must be excepted; we then have 1 — cos a = 0; that isto say, the arcs u and v are the same. In this case the term\ cosy Qj, + v)q still gives a development whose sum is nothing;

SECT. II.] ELIMINATION. 251

but the quantity \ cosj (fi— v) q furnishes equal terms, each ofwhich has the value \ ; hence the sum of the products term byterm of the two first series is | n .

In the same manner we can find the value of the sum of theproducts term by term of the two second series, or

2 (cos j / ^ cos j * ^ ) ;

in fact, we can substitute for cos j^q cos jvq the quantity

i cosj (ji - v) q + £ cosj (ft + v) q,

and we then conclude, as in the preceding case, that S^cos j(/J>+v)qis nothing, and that S^ cosj (v< — v)q is nothing, except in the casewhere //. = i>. It follows from this that the sum of the productsterm by term of the two second series, or 2 (cos j/xq cos jvq), isalways 0 when the arcs u and v are different, and equal to \nwhen u = v. It only remains to notice the case in which the arcsfiq and vq are both nothing, when we have 0 as the value of

t (smjfiq sin jvq),

which denotes the sum of the products term by term of the twofirst series.

The same is not the case with the sum ^ ( c o s j ^ cos jvq) takenwhen jxq and vq are both nothing; the sum of the products termby term of the two second series is evidently equal to n.

As to the sum of the products term by term of the two series


cos Ow, cos lu, cos 2u, cos 3u, ... cos (n — 1) u,

it is nothing in all cases, as may easily be ascertained by the fore-going analysis.

270. The comparison then of these series furnishes the follow-ing results. If we divide the circumference 2ir into n equalparts, and take an arc u composed of an integral number fi ofthese parts, and mark the ends of the arcs u, 2u, 3u,... (n— l)u, itfollows from the known properties of trigonometrical quantitiesthat the quantities

sin Ow, sin lu, sin 2u, sin 3M, ... sin (n — 1) u,


or indeed

cos 0u} cos lu, co$2i/, cos 3u, ... cos (n- 1) u,

form a recurring periodic series composed of n terms: if we Corn-er

pare one of the two series corresponding to an arc u or /A —

with a series corresponding to another arc v or v —, and

multiply term by term the two compared series, the sum of theproducts will be nothing when the arcs u and v are different. Ifthe arcs u and v are equal, the sum of the products is equal to Jw,when we combine two series of sines, or when we combine twoseries of cosines; but the sum is nothing if we combine a series ofsines with a series of cosines. If we suppose the arcs u and v tobe nul, it is evident that the sum of the products term by term isnothing whenever one of the two series is formed of sines, or whenboth are so formed, but the sum of the products is n if the com-bined series both consist of cosines. In general, the sum of theproducts term by term is equal to 0, or \n or n\ known formulaewould, moreover, lead directly to the same results. They are pro-duced here as evident consequences of elementary theorems intrigonometry.

271. By means of these remarks it is easy to effect the elimi-nation of the unknowns in the preceding equations. The unknownAx disappears of itself through having nul coefficients; to find Bx

we must multiply the two members of each equation by the co-efficient of JB1 in that equation, and on adding all the equationsthus multiplied, we find

To determine A2 we must multiply the two members of eachequation by the coefficient of A2 in that equation, and denoting

the arc — by q} we have, after adding the equations together,

ax sin Oq + a2 sin \q + a3 sin 2q + ... + an sin (n-l)q = - n A2.

Similarly to determine B2 we have

ax cos Oq + a2 cos \q + an cos 2q -f ... + an cos (n - 1) q = ~ n B%.

SECT. II.] VALUE OF THE COEFFICIENTS. 253

In general we could find each unknown by multiplying thetwo members of each equation by the coefficient of the unknownin that equation, and adding the products. Thus we arrive at thefollowing results:

71 A • n2<7r . , 2TT . O2TT D ~, . , . - . - 2 T T-A = a s i n O — + a Q s i n l — - f a q s i n 2 — +&c. = i a i s i n ( i — 1 ) 1 — ,

^ . B ^ a - c o s O — + a 0 c o s l — -}-aqcos2— +&c.= S« , cos ( i—1)1—,^ x 7i 2 ft 3 ft v / n

- A = a sin 0.2 h a9 sinl. 2 f- aqsin 2.2 h&c = 2a, sin (t — 1) 2 — ,

- 5 = a,cos 0.2 Y a9cosl.2 —haqcos2.2 —j-&c.=2a,-cos (i —1)2 —,

^ 4 = a l S i n 0 . 3 — + a a s i n l . 3 — +a 3 s in2 .3—+&c.=2a , s in ( i - l )3 — ,

- 5 , = ^008 0.3 —ha2cosl .3 —haqcos2.3 —h&c. = 2a,cos(i —1)3 — 9

&c (M).

To find the development indicated by the symbol 2, we mustgive to i its n successive values 1, 2, 3, 4, &c, and take the sum,in which case we have in general

-As=%ai$m(i —1)0'—1)— and ^B =2aiCos(i—l)(j—1)^— -

If we give to the integer^" all the successive values 1, 2, 3, 4,&c. which it can take, the two formulae give our equations, and ifwe develope the term under the sign 2, by giving to i its n values1, 2, 3, ... n, we have the.values of the unknowns A1,B1, A2,B2,Az, B3, &c, and the equations (m), Art. 267, are completely solved.

272. We now substitute the known values of the coefficientsAv Blf A2,B2, A^B^ &c.; in equations (ji)9 Art. 266, and obtainthe following values :

ax = NQ + iV; e* ver8in * + N2€ *versin Q* + &c.

a2 = JSrQ + {Mt sin q1 + Nx cos qt) e'vers in*

+ {M2 sin £2 + N2 cos g2) e'versin<^ + &c.

a3 = iV0 + (Jf l sin 2 ^ + iVA cos 2^) e *versin ffi

+ (M2 sin 2g2 + i^2 cos 2q2) ef versin 9* + &c.

a, = iST0+ {JflSin ( i - 1) ? 1 + JV; cos( j - 1) ?1} 6i v e r s i n ^

2 sin (j - 1) & + i^2 cos ( j - 1) g2} e«versin * + &c.

an = NQ 4- { ^ sin (w - 1) qx + Nt cos (n - 1) ^ } € ' v e r 8 i n^

+ [M2 sin (w - 1) q2 + N2 cos (n - 1) #2} e *versin «»+ &c.

In these equations

-™ , 2TT O 2 T T Q 2 T T

2 2iV^-S^COS (^-1) Jj, Jlfj = -

2 2iV2 = - 2a»cos (i - 1) gra, M2 = - 2 a , sin (i - 1) g2,

2 2i ^ = - 2a f cos (i - 1) q3, M3 = - 2a< sin (z - 1 ) qy

&C. &C.

273. The equations which we have just set down contain thecomplete solution of the proposed problem; it is represented bythe general equation

1 ^ T2 . . 1 N 2 T T ^ . , . 1 X2TTa i = - 2 « i + - s m ( j - 1) — 2 a < s m ^ — 1) —

+ - cos (j — 1) — Sa» cos (z — 1) — \

_ _ _ _ m n

n(6),

SECT. II.J APPLICATION OF THE SOLUTION. 255

in which only known quantities enter, namely, av a2, a3 ... an,which are the initial temperatures, k the measure of the con-ducibility, m the value of the mass, n the number of massesheated, and t the time elapsed.

From the foregoing analysis it follows, that if several equalbodies n in number are arranged in a circle, and, having receivedany initial temperatures, begin to communicate heat to each otherin the manner we have supposed; the mass, of each body beingdenoted by m, the time by t, and a certain constant coefficient byh, the variable temperature of each mass, which must be a functionof the quantities t, m, and Jc, and of all the initial temperatures,is given by the general equation (e). We first substitute insteadof j the number which indicates the place of the body whosetemperature we wish to ascertain, that is to say, 1 for the firstbody, 2 for the second, &c; then with respect to the letter i whichenters under the sign 2, we give to it the n successive values1, 2, 3, ... n, and take the sum of all the terms. As to thenumber of terms which enter into this equation, there must beas many of them as there are different versed sines belonging tothe successive arcs

n n n n

that is to say, whether the number n be equal to (2\ + 1) or 2A,,according as it is odd or even, the number of terms which enterinto the general equation is always \ + 1.

274. To give an example of the application of this formula,let us suppose that the first mass is the only one which at firstwas heated, so that the initial temperatures av a2, a3 ...an are allnul, except the first. It is evident that the quantity of heatcontained in the first mass is distributed gradually among all theothers. Hence the law of the communication of heat is expressedby the equation

a i = ~ai + ~ ai cos (7 — 1 ) — e m

2 f. I N O 2 T T -2-+ - ^ 0 0 3 ( 7 — 1)2 — e

+ - a , cos (7 —1) 3 — e m n + &c.


If the second mass alone had been heated and the tempera-tures av as, av ... an were nul, we should have

1 2 f . , . 1N2?r . 2TTaj ~ ~ a2 + "" a2 )S111 \J "" -*"/ S111

+ cos ( / — I ) — c o s — [ e~~>»™imi1'"w 71 % J

2 ( . , . n N o 2TT . O 2 T T+ - a9 1 sm (7 — 1) 2 — sm 2 —

+ cos 0 - 1 ) 2 — cos 2 —

&c,

and if all the initial temperatures were supposed nul, excepta1 and a2, we should find for the value of a} the sum of the valuesfound in each of the two preceding hypotheses. In general it iseasy to conclude from the general equation (e), Art. 273, that inorder to find the law according to which the initial quantities ofheat are distributed between the masses, we may consider sepa-rately the cases in which the initial temperatures are nul, one onlyexcepted. The quantity of heat contained in one of the massesmay be supposed to communicate itself to all the others, regardingthe latter as affected with nul temperatures; and having madethis hypothesis for each particular mass with respect to the initialheat which it has received, we can ascertain the temperature ofany one of the bodies, after a given time, by adding all thetemperatures which the same body ought to have received oneach of the foregoing hypotheses.

275. If in the general equation (e) which gives the value of

0Ld> we suppose the time to be infinite, we find a3 = - 2 au so thatIft

each of the masses has acquired the mean temperature; a resultwhich is self-evident.

As the value of the time increases, the first term - %a{

nbecomes greater and greater relatively to the following terms, orto their sum. The same is the case with the second with respectto the terms which follow it; and, when the time has become

SECT. II.] LATER TEMPERATURES. 2 5 7

considerable, the value of a,- is represented without sensible errorby the equation,

l v 2 f . , . - v 2 T T ^ . . . -, v 2 7 T

«> = -2a, + - j smO- l )—2a mn(*-l) —

+ cos 0 - 1 ) — 2aicos(*- 1) —[2TT) -™ versing

6 w w.

Denoting by a and 6 the coefficients of sin (j — 1) — and of

cos (j — 1) —, and the fraction e M w by G>, we nave

1 ^ f . , . -.N2TT 7 , . - x 27rCL- = - ia* + ]asin 17 — 1) — + 6 cos (9 — 1) —

The quantities a and Z> are constant, that is to say, independentof the time and of the letter j which indicates the order of themass whose variable temperature is ajt These quantities are thesame for all the masses. The difference of the variable tempera-ture otj from the final temperature - ^at decreases therefore for

each of the masses, in proportion to the successive powers of thefraction co. Each of the bodies tends more and more to acquire

the final temperature - S ai3 and the difference between that

final limit and the variable temperature of the same body endsalways by decreasing according to the successive powers of afraction. This fraction is the same, whatever be the body whosechanges of temperature are considered; the coefficient of co* or

(a sin Uj + b cos Uj), denoting by Uj the arc (j — 1) — , may be put

under the form A sin (uj 4- B), taking A and B so as to havea = A cos B, and b = A sin B. If we wish to determine thecoefficient of co* with regard to the successive bodies whosetemperature is oi+1, ai+2, ai+3, &c, we must add to Uj the arc

— or 2 — , and so on; so that we have the equationsn n

<Xj — Xcii = A sin (B + uj) G>* + &c.

or,- - - Xai = A sin [B + Uj + 1 —+1 n \ n

F. H. 17

a +i- -1 %a{ = A sin (B + nj + 2 — ) <of + &c.

a3+,-^tai = A sin (B + UJ + 3 ^ co*+ &c.

&c.276. We see, by these equations, that the later differences

between the actual temperatures and the final temperatures arerepresented by the preceding equations, preserving only the firstterm of the second member of each equation. These later differ-ences vary then according to the following law: if we consideronly one body, the variable difference in question, that is to say?

the excess of the actual temperature of the body over the finaland common temperature, diminishes according to the successivepowers of a fraction, as the time increases by equal parts; and, ifwe compare at the same instant the temperatures of all thebodies, the difference in question varies proportionally to the suc-cessive sines of the circumference divided into equal parts. Thetemperature of the same bbdy, taken at different successive equalinstants, is represented by the ordinates of a logarithmic curve,whose axis is divided into equal parts, and the temperature ofeach of these bodies, taken at the same instant for all, is repre-sented by the ordinates of a circle whose circumference is dividedinto equal parts. It is easy to see, as we have remarked before,that if the initial temperatures are such, that the differences ofthese temperatures from the mean or final temperature are pro-portional to the successive sines of multiple arcs, these differenceswill all diminish at the same time without ceasing to be propor-tional to the same sines. This law, which governs also the initialtemperatures, will not be disturbed by the reciprocal action of thebodies, and will be maintained until they have all acquired acommon temperature. The difference will diminish for each bodyaccording to the successive powers of the same fraction. Such isthe simplest law to which the communication of heat between asuccession of equal masses can be submitted. When this law hasonce been established between the initial temperatures, it is main-tained of itself; and when it does not govern the initial tempera-tures, that is to say, when the differences of these temperaturesfrom the mean temperature are not proportional to successivesines of multiple arcs, the law in question tends always to be set

SECT. II.] CONTINUOUS MASSES IN A RING. 259

up, and the system of variable temperatures ends soon by coin-ciding sensibly with that which depends on the ordinates of acircle and those of a logarithmic curve.

Since the later differences between the excess of the tempera-ture of a body over the mean temperature are proportional tothe sine of the arc at the end of which the body is placed, itfollows that if we regard two bodies situated at the ends of thesame diameter, the temperature of the first will surpass the meanand constant temperature as much as that constant temperaturesurpasses the temperature of the second body. For this reason, ifwe take at each instant the sum of the temperatures of twomasses whose situation is opposite, we find a constant sum, andthis sum has the same value for any two masses situated at theends of the same diameter.

277. The formulae which represent the variable temperaturesof separate masses are easily applied to the propagation of heatin continuous bodies. To give a remarkable example, we willdetermine the movement of heat in a ring, by means of thegeneral equation which has been already set down.

Let it be supposed that n the number of masses increases suc-cessively, and that at the same time the length of each massdecreases in the same ratio, so that the length of the system hasa constant value equal to 2TT. Thus if n the number of massesbe successively 2, 4, 8, 16, to infinity, each of the masses will

7T 7T 7T

be 7r, —; —, - , &c. It must also be assumed that thefacility with which heat is transmitted increases in the sameratio as the number of masses m; thus the quantity which krepresents when there are only two masses becomes double whenthere are four, quadruple when there are eight, and so on.Denoting this quantity by g, we see that the number k must besuccessively replaced by g, 2g, 4<g, &c. If we pass now to thehypothesis of a continuous body, we must write instead of m, thevalue of each infinitely small mass, the element dx; instead of n,

the number of masses, we must write y-: instead of k writeax

n irq

17—2

As to the initial temperatures av a2, ar..an) they depend onthe value of the arc x, and regarding these temperatures as thesuccessive states of the same variable, the general value at repre-sents an arbitrary function of x. The index i must then be

replaced by-T-. With respect to the quantities <xv a2, aa, ...,

these are variable temperatures depending on two quantitiesx and t Denoting the variable by v9 we have v = </> (a?, t). Theindex j , which marks the place occupied by one of the bodies,

rv*

should be replaced by -^-. Thus, to apply the previous analysis to

the case of an infinite number of layers, forming a continuousbody in the form of a ring, we must substitute for the quanti-ties n, m, k, civ i, aj} j , their corresponding quantities, namely,

-7— > dx} ~r- > f(x)> ~r~» <£ ix> 0? "J~ • Ije^ these substitutions be

made in equation (e) Art. 273, and let ^ da? be written instead

of versin dx, and i and j instead of i — 1 and j — 1. The first

term - 2a< becomes the value of the integral — \f(x) dx taken from

# = 0 to # = 2TT; the quantity sin (j - 1) ?Z[ becomes sin jdx orn

sin x; the value of cos (J— 1) -j- is cos x; that of - 2 ^ sin (i — 1) —ax n K J n

is - I f{x) sin xdoo, the integral being taken from x = 0 to X=2TT\

and the value of - Xa c^s (i - 1) — is - I f(x) cos x dx, then n TTJ J J

integral being taken betw een the same limits. Thus we obtainthe equation

+ - ( sin x I /(x) sin xdx + cos x If {x) cos xdx je-ff*

jf(x) cos 2aj dx) e~

SECT. II.] REMARKS, 261

and representing the quantity grtr by fc, we have

7TV = - \f(x)dx+[ sina? l/(^) sina?cZx+cosccl/(^) cos^cicle w

4- (sin 2x \f{x) sin 2^dx+cos 2xjf(x) cos 2# da?J 6 " *2/a

+ &c.

278. This solution is the same as that which was given in thepreceding section, Art. 241; it gives rise to several remarks. 1st.It is not necessary to resort to the analysis of partial differentialequations in order to obtain the general equation which expressesthe movement of heat in a ring. The problem may be solved fora definite number of bodies, and that number may then be sup-posed infinite. This method has a clearness peculiar to itself, andguides our first researches. It is easy afterwards to pass to amore concise method by a process indicated naturally. We seethat the discrimination of the particular values, which, satisfyingthe partial differential equation, compose the general value, isderived from the known rule for the integration of linear differ-ential equations whose coefficients are constant. The discrimina-tion is moreover founded, as we have seen above, on the physicalconditions of the problem. 2nd. To pass from the case of separatemasses to that of a continuous body, we supposed the coefficient kto be increased in proportion to n, the number of masses. Thiscontinual change of the number k follows from what we haveformerly proved, namely, that the quantity of heat which flowsbetween two layers of the same prism is proportional to the value

civof -j-, x denoting the abscissa which corresponds to the section,

and v the temperature. If, indeed, we did not suppose the co-efficient k to increase in proportion to the number of masses, butwere to retain a constant value for that coefficient, we shouldfind, on making n infinite, a result contrary to that which isobserved in continuous bodies. The diffusion of heat would beinfinitely slow, and in whatever manner the mass was heated, thetemperature at a point would suffer no sensible change duringa finite time, which is contrary to fact. Whenever we resort tothe consideration of an infinite num'ber of separate masses which

transmit heat, and wish to pass to the case of continuous bodies,we must attribute to the coefficient k, which measures the velocityof transmission, a value proportional to the number of infinitelysmall masses which compose the given body.

3rd. If in the last equation which we obtained to express thevalue of v or <£ (x, t), we suppose t = 0, the equation necessarilyrepresents the initial state, we have therefore in this way theequation (p), which we obtained formerly in Art. 233, namely,

+s in# I f(x)sinxdx+sin2xl f(x)sm2xdx + &c.

+ cos x \ f{x)Q,osxdx-\-Q,os2x I f(x) cos 2xdx + &c»

Thus the theorem which gives, between assigned limits, thedevelopment of an arbitrary function in a series of sines or cosinesof multiple arcs is deduced from elementary rules of analysis.Here we find the origin of the process which we employed tomake all the coefficients except one disappear by successive in-tegrations from the equation

, , N -f a, sin x + ao sin 2% + ao sin Sx + &c.K ' + b1 cos x 4- b^ cos 2x + b3 cos Sx 4- &c.

These integrations correspond to the elimination of the differentunknowns in equations (m), Arts. 267 and 271, and we see clearlyby the comparison of the two methods, that equation (B), Art. 279,holds for all values of x included between 0 and 2IT, without itsbeing established so as to apply to values of x which exceed thoselimits.

279. The function <f> (x} t) which satisfies the conditions ofthe problem, and whose value is determined by equation (E),Art. 277, may be expressed as follows:

) = jdaf((x) + {2 sinx I dzf(a) sma + 2cos-#lda/'(a)cosa \e-u

{2sin3a5 f^a/(a)siu3a4-2cos3^f^/(a)cos3a}^32^+ &c.

SECT. II.] FUNCTIONAL EXPRESSION. 263

or 2ir(f>(xi t) = I rfa/(a) {1 + (2 sin x sin a + 2 cos a? cos a) <T**

+ (2 sin 2x sin 2a + 2 cos 2-c cos 2oL)e"mt

+ (2 sin 3«c sin 3a + 2 cos Zx cos 3a) e~^u + &c.}

= jd.f(o) [1 + 22 cos i (a - x) e~im\

The sign 2 affects the number i, and indicates that the summust be taken from i—l to i = oo. We can also include thefirst term under the sign 2, and we have

= Ida/(a) 2 ! I cos * (a - a?)a?,

We must then give to i all integral values from — co to 4- oo ;which is indicated by writing the limits — oo and + oo next to thesign 2, one of these values of % being 0. This is the most conciseexpression of the solution. To develope the second member of theequation, we suppose i=0, and then i~ 1, 2, 3, &c, and doubleeach result except the first, which corresponds to i = 0. Whent is nothing, the function cf> (x, t) necessarily represents the initialstate in which the temperatures are equal to f(x), we have there-fore the identical equation,

-a:) (B).

We have attached to the signs I aud 2 the limits between

which the integral sum must be taken. This theorem holdsgenerally whatever be the form of the function / (x) in the in-terval from x = 0 to x = 2ir; the same is the case with that whichis expressed by the equations which give the development of F (x),Art. 235; and we shall see in the sequel that we can prove directlythe truth of equation (B) independently of the foregoing con-siderations.

280. It is easy to see that the problem admits of no solutiondifferent from that given by equation (E), Art. 277. The function</> (x, t) in fact completely satisfied the conditions of the problem,

and from the nature of the differential equation ~ = h -~2, nodo (tX

other function can enjoy the same property. To convince our-selves of this we must consider that when the first state of the

-~solid is represented by a given equation v1=f(x)9 the fluxion

d? fix)is known, since it is equivalent to k J, \ '•. Thus denoting by

v2 or vx + Jc-jf dt, the temperature at the commencement of theCtt/

second instant, we can deduce the value of v2 from the initialstate and from the differential equation. We could ascertain inthe same manner the values v3) vv ... vn of the temperature atany point whatever of the solid at the beginning of each instant.Now the function (j> (x, t) satisfies the initial state, since we have(f> (x, 0) =/ (#) . Further, it satisfies also the differential equation;consequently if it were differentiated, it would give the same

values for -^r > i f , -JT> &c., as would result from successivedt dt dt '

applications of the differential equation (a). Hence, if in thefunction <£> (x, t) we give to t successively the values 0, co, 2eu,3&>, &c, co denoting an element of time, we shall find the samevalues vx, v2, vs, &c. as we could have derived from the initial

state by continued application of the equation -j- =^-^—2. Hencectt dec

every function tjr (#, t) which satisfies the differential equation andthe initial state necessarily coincides with the function tf>(%,t):for such functions each give the same function of x, when in themwe suppose t successively equal to 0, co, 2&), 3eo ... ico, &c.

We see by this that there can be only one solution of theproblem, and that if we discover in any manner a function yjr (xt t)which satisfies the differential equation and the initial state, weare certain that it is the same as the former function given byequation (E).

281. The same remark applies to all investigations whoseobject is the varied movement of heat; it follows evidently fromthe very form of the general equation.

For the same reason the integral of the equation -r- = h j - 2

can contain only one arbitrary function of x. In fact, when a

SECT. II.] GENERAL INTEGRAL. 265

value of v as a function of x is assigned for a certain value ofthe time t, it is evident that all the other values of v whichcorrespond to any time whatever are determinate. We maytherefore select arbitrarily the function of x, which correspondsto a certain state, and the general function of the two variablesx and t then becomes determined. The same is not the case

with the equation -1-2 +-r-g = 0, which was employed in the

preceding chapter, and which belongs to the constant movementof heat; its integral contains two arbitrary functions of x and y:but we may reduce this investigation to that of the varied move-ment, by regarding the final and permanent state as derived fromthe states which precede it, and consequently from the initialstate, which is given.

The integral which we have given

t — x)

contains one arbitrary function f(x)y and has the same extent asthe general integral, which also contains only one arbitrary func-tion of x; or rather, it is this integral itself arranged in a formsuitable to the problem. In fact, the equation vx =f (x) represent-ing the initial state, and v = cf> (x, t) representing the variablestate which succeeds it, we see from the very form of the heatedsolid that the value of v does not change when x ± i2ir is writteninstead of x, i being any positive integer. The function

— \daf(a) ze~{2kt cos i (a — x)

satisfies this condition; it represents also the initial state whenwe suppose t = 0, since we then have

/(#) = ^7r]dj'f (°0 2 cos i (a - x)y

an equation which was proved above, Arts. 235 and 279, and is

also easily verified. Lastly, the same function satisfies the differ-

ential equation j = k - p ' . Whatever be the value of t, the

temperature v is given by a very convergent series, and the differentterms represent all the partial movements which combine to form

266 THEOKY OF HEAT. [CHAP. IV.

the total movement. As the time increases, the partial states ofhigher orders alter rapidly, but their influence becomes inappre-ciable; so that the number of values which ought to be given tothe exponent i diminishes continually. After a certain time thesystem of temperatures is represented sensibly by the terms whichare found on giving to i the values 0, ,± 1 and ± 2, or only 0

and + 1, or lastly, by the first of those terms, namely, ~— I dzf (a);

there is therefore a manifest relation between the form of thesolution and the progress of the physical phenomenon which hasbeen submitted to analysis.

282. To arrive at the solution we considered first the simplevalues of the function v which satisfy the differential equation:we then formed a value which agrees with, the initial state, andhas consequently all the generality which belongs to the problem.We might follow a different course, and- derive the same solutionfrom another expression of the integral; when once the solutionis known, the results are easily transformed. If we suppose thediameter of the mean section of the ring to increase infinitely, thefunction <j> (x, t), as we shall see in the sequel, receives a differentform, and coincides with an integral which contains a singlearbitrary function under the sign of the definite integral. Thelatter integral might also be applied to the actual problem; but,if we were limited to this application, we should have but a veryimperfect knowledge of the phenomenon; for the values of thetemperatures would not be expressed by convergent series, andwe could not discriminate between the states which succeed eachother as the time increases. The periodic form which the problemsupposes must therefore be attributed to the function which re-presents the initial state; but on modifying that integral in thismanner, we should obtain no other result than

</> (x, i) = -^fdoif (a) Se-*1*4 cos i (a - ,r).

From the last equation we pass easily to the integral inquestion, as was proved in the memoir which preceded this work.It is not less easy to obtain the equation from the integral itself.These transformations make the agreement of the analytical.results more clearly evident; but they add nothing to the theory,

SECT. II.J DIFFERENT INTEGRAL FORMS. 267

and constitute no different analysis. In one of the followingchapters we shall examine the different forms which may be

dv d2vassumed by the integral of the equation -rr — k~T~2> the relations

which they have to each other, and the cases in which they oughtto be employed.

To form the integral which expresses the movement of heat ina ring, it was necessary to resolve an arbitrary function into aseries of sines and cosines of multiple arcs; the numbers whichaffect the variable under the symbols sine and cosine are thenatural numbers 1, 2, 3, 4, &c. In the following problem thearbitrary function is again reduced to a series of sines; but thecoefficients of the variable under the symbol sine are no longerthe numbers 1, 2, 3, 4, &c: these coefficients satisfy a definiteequation whose roots are all incommensurable and infinite innumber.

Note on Sect. I, Chap. IV. Guglielmo Libri of Florence was the first toinvestigate the problem of the movement of heat in a ring on the hypothesis ofthe law of cooling established by Dulong and Petit. See his Memoire sur latheorie de la chaleur, Crelle's Journal, Band VII., pp. 116—131, Berlin, 1831.(Read before the French Academy of Sciences, 1825.) M. Libri made the solutiondepend upon a series of partial differential equations, treating them as if theywere linear. The equations have been discussed in a different manner byMr Kelland, in his Theory of Heat, pp. 69—75, Cambridge, 1837. The principalresult obtained is that the mean of the temperatures at opposite ends of anydiameter of the ring is the same at the same instant. [A. F.]

CHAPTER V.

OF THE PROPAGATION OF HEAT IN A SOLID SPHERE.

SECTION I.

General solution.

283. THE problem of the propagation of heat in a sphere hasbeen explained in Chapter II., Section 2, Article 117; it consistsin integrating the equation

dv _l rd2v 2 dvJt~~ \ d Y + d

so that when x = X the integral may satisfy the condition

dx

k denoting the ratio - ^ , and h the ratio -^ of the two con-

ducibilities; v is the temperature which is observed afterthe time t has elapsed in a spherical layer whose radius is x\X is the radius of the sphere; v is a function of x and t, which isequal to F(x) when we suppose £ = 0. The function F{x) isgiven, and represents the initial and arbitrary state of the solid.

If we make y = vx, y being a new unknown, we have,dti d^t/

after the substitutions, ^7~^3"^ : t n u s w e must integrate the

last equation, and then take v = -. We shall examine, in the

first place, what are the simplest values which can be attributedto yt and then form a general value which will satisfy at the same

CHAP. V. SECT. I.] PAKTICULAB SOLUTIONS. 269

time the differential equation, the condition relative to thesurface, and the initial state. It is easily seen that when thesethree conditions are fulfilled, the solution is complete, and noother can be found.

284. Let y = emtu, u.being a function of x, we have

d2u

First, we notice that when the value of t becomes infinite, thevalue of v must be nothing at all points, since the body is com-pletely cooled. Negative values only can therefore be taken form. Now Jc has a positive numerical value, hence we concludethat the value of u is a circular function, which follows from theknown nature of the equation

7 d\dx2

Let u = A cos nx + B sin nx; we have the condition m = — kn2.Thus we can express a particular value of v by the equation

Q-TcnH

v = (A cos nx + B sin nx),XX

where n is any positive number, and A and B are constants. Wemay remark, first, that the constant A ought to be nothing; forthe value of v which expresses the temperature at the centre,when we make x = 0, cannot be infinite; hence the term A cos nxshould be omitted.

Further, the number n cannot be taken arbitrarily. In fact,/if}

if in the definite equation -y- + ho = 0 we substitute the value

of v, we findnx cos nx + (hx — 1) sin nx = 0.

As the equation ought to hold at the surface, we shall supposein it x = X the radius of the sphere, which gives

Let X be the number 1 — hX, and nX= e, we have = \tane

We must therefore find an arc e, which divided by its tangent

270 THEORY OF HEAT. [CHAP. V.

isgives a known quotient \, and afterwards take n==~y-

evident that there are an infinity of such arcs, which have a givenratio to their tangent; so that the equation of condition

nX -tan nX

has an infinite number of real roots.

285. Graphical constructions are very suitable for exhibitingthe nature of this equation. Let u - tan e (fig. 12), be the equation

Fig. 12.

to a curve, of which the arc e is the abscissa, and u the ordinate;

and let u = - be the equation to a straight line, whose co-ordinates

are also denoted by e and u. If we eliminate u from these two

equations, we have the proposed equation - = tan e. The un-

known e is therefore the abscissa of the point of intersection ofthe curve and the straight line. This curved line is composed ofan infinity of arcs; all the ordinates corresponding to abscissae

1 3 5 7 ,217"9 27F' I77"' 271"'

are infinite, and all those which correspond to the points oy TT,2-TT, 3-7T, &c. are nothing. To trace the straight line whose

equation is u = -= . we form the square oicoi, and

measuring the quantity hX from co to h, join the point h withthe origin a The curve non whose equation is u — tan e hns for

SECT. I.] ROOTS OF EQUATION OF CONDITION. 271

tangent at the origin a line which divides the right angle into twoequal parts, since the ultimate ratio of the arc to the tangent is 1.We conclude from this that if X or Z — hX is a quantity less thanunity, the straight line mom passes from the origin above thecurve non, and there is a point of intersection of the straight linewith the first branch. It is equally clear that the same straightline cuts all the further branches nun, n^irn, &c. Hence the

equation = \ has an infinite number of real roots. Thetan €

77*first is included between 0 and —, the second between ir and

Zi

—, the third between 2ir and -^-, and so on. These roots

approach very near to their upper limits when they are of a veryadvanced order.

286. If we wish to calculate the value of one of the roots,for example, of the first, we may employ the following rule : write

down the two equations e = arc tan u and u = - , arc tan u de-Ay

noting the length of the arc whose tangent is u. Then takingany number for u, deduce from the first equation the value of e;substitute this value in the second equation, and deduce anothervalue of u; substitute the second value of u in the first equation;thence we deduce a value of e, which, by means of the secondequation, gives a third value of u. Substituting it in the firstequation we have a new value of e. Continue thus to determineu by the second equation, and e by the first. The operation givesvalues more and more nearly approaching to the unknown e, as isevident from the following construction.

In fact, if the point u correspond (see fig. 13) to the arbitraryvalue which is assigned to the ordinate u; and if we substitutethis value in the first equation € = arc tan u, the point e willcorrespond to the abscissa which we have calculated by meansof this equation. If this abscissa e be substituted in the second

equation u = - , we shall find an ordinate u which corresponds

to the point u. Substituting u in the first equation, we find anabscissa e which corresponds to the point e'; this abscissa being


then substituted in the second equation gives rise to an ordinatewhich when substituted in the first, gives rise to a thirdu

abscissa e", and so on to infinity. That is to say, in order torepresent the continued alternate employment of the two

Fig. 13. Fig. 14.

ceding equations, we must draw through the point u a horizontalline up to the curve, and through e the point of intersection drawa vertical as far as the straight line, through the point of inter-section u draw a horizontal up to the curve, through the point ofintersection e draw a vertical as far as the straight line, and so onto infinity, descending more and more towards the point sought.

287. The foregoing figure (13) represents the case in whichthe ordinate arbitrarily chosen for u is greater than that whichcorresponds to the point of intersection. If, on the other hand, wechose for the initial value of u a smaller quantity, and employed

in the same manner the two equations e = arc tan u> u = - , weA

should again arrive at values successively closer to the unknownvalue. Figure 14 shews that in this case we rise continuallytowards the point of intersection by passing through the pointsueu € u" e", &c. which terminate the horizontal and vertical lines.Starting from a value of u which is too small, we obtain quantitiese e e" e"', &c. which converge towards the unknown value, and aresmaller than it; and starting from a value of u which is too great,we obtain quantities which also converge to the unknown value,and each of which is greater than it. We therefore ascertain

SECT. I.] MODE OF APPROXIMATION. 273

successively closer limits between the which magnitude sought isalways included. Either approximation is represented by theformula

€ = arc tan - arc tan \- arc tan ( - arc tan - ) M[_A, (A, \ \ ^/JJ

When several of the operations indicated have been effected,the successive results differ less and less, and we have arrived atan approximate value of e.

288. We might attempt to apply the two equations

€ = arc tan u and u = -

in a different order, giving them the form w = tane and e = Xu.We should then take an arbitrary value of e, and, substituting itin the first equation, we should find a value of u, which beingsubstituted in the second equation would give a second value ofe; this new value of e could then be employed in the samemanner as the first. But it is easy to see, by the constructionsof the figures, that in following this course of operations wedepart more and more from the point of intersection instead ofapproaching it, as in the former case. The successive values of ewhich we should obtain would diminish continually to zero, orwould increase without limit. We should pass successively frome" to u", from u" to e', from e to u, from u! to e, and so on toinfinity.

The rule which we have just explained being applicable to thecalculation of each of the roots of the equation

tan 6

which moreover have given limits, we must regard all these rootsas known numbers. Otherwise, it was only necessary to be as-sured that the equation has an infinite number of real roots.We have explained this process of approximation because it isfounded on a remarkable construction, which may be usefullyemployed in several cases, and which exhibits immediately thenature and limits of the roots; but the actual application of theprocess to the equation in question would be tedious; it would beeasy to resort in practice to some other mode of approximation.

F. H. 18


289. We now know a particular form which may be given tothe function v so as to satisfy the two conditions of the problem.This solution is represented by the equation

Ae~knH sin noc , 2/ sin nxv — or v = ae~kn l .

x nx

The coefficient a is any number whatever, and the number n isnX

such that ± ^ = 1 — hX. It follows from this that if thetanwJL

initial temperatures of the different layers were proportional toQ"l ~V"\ /Yj '"V*

the quotient , they would all diminish together, retainingfix

between themselves throughout the whole duration of the coolingthe ratios which had been set up ; and the temperature at eachpoint would decrease as the ordinate of a logarithmic curve whoseabscissa would denote the time passed. Suppose, then, the arc ebeing divided into equal parts and taken as abscissa, we raise ateach point of division an ordinate equal to the ratio of the sine tothe arc. The system of ordinates will indicate the initial tem-peratures, which must be assigned to the different layers, from thecentre to the surface, the whole radius X being divided into equalparts. The arc e which, on this construction, represents theradius X, cannot be taken arbitrarily; it is necessary that thearc and its tangent should be in a given ratio. As there arean infinite number of arcs which satisfy this condition, we mightthus form an infinite number of systems of initial temperatures,which could exist of themselves in the sphere, without the ratiosof the temperatures changing during the cooling.

290. It remains only to form any initial state by means ofa certain number, or of an infinite number of partial states, eachof which represents one of the systems of temperatures which wehave recently considered, in which the ordinate varies with thedistance x} and is proportional to the quotient of the sine by thearc. The general movement of heat in the interior of a spherewill then be decomposed into so many particular movements, eachof which is accomplished freely, as if it alone existed.

Denoting by nv ?i2, ra3, &c, the quantities which satisfy the

equation -^= 1 -hX, and supposing them to be arranged in

SECT. I.] COEFFICIENTS OF THE SOLUTION. 275

order, beginning with the least, we form the general equa-tion

vx = a^"^2' sin nxx + a2e~^2' sin i\x + a3e~kn*H sin n3x + &c.

If t be made equal to 0, we have as the expression of theinitial state of temperatures

vx = ax sin nxx + a2 sin n2x + a3 sin n3x + &c.

The problem consists in determining the coefficients aiy a2, a3

&c, whatever be the initial state. Suppose then that we knowthe values of v from x = 0 to x = X, and represent this system ofvalues by F(x); we have

F(x) = - (ax sin nxx + ct2 sin n%x + a3 sin n3x + a4 sin n^x + &c.)\.. (e).x

291. To determine the coefficient al9 multiply both membersof the equation by x sin nx dx, and integrate from x = 0 to x = X.

The integral I s i nw^s innx^ taken between these limits is

TZ (— m sin nXcos mX+ ft sin mXcos wX).m —nIf m and w are numbers chosen from the roots nv n2, n3,

y

&e., which satisfy the equation ^ = 1 - hX, we havetan Tt JL

mX nXtan mX tan nXy

or m cos mXsin nX— n sin mXcos nX = 0.

We see by this that the whole value of the integral is nothing;but a single case exists in which the integral does not vanish,

namely, when m = n. It then becomes j : ; and, by application of

known rules, is reduced to

lx-±sin2nX.2 in

1 Of the possibility of representing an arbitrary function by a series of thisform a demonstration has been given by Sir W. Thomson, Camb, Math. Journal,Vol. in. pp. 25—27. [A. F.]

18—2

It follows from this that in order to obtain the value of thecoefficient alf in equation (e), we must write

2 jx sinnxxF(x) dx = ax [X — x— sin2nxXj ,

the integral being taken from x = 0 to x = X. Similarly we have

2 jx sin n2x F(x) dx = aJX — ^—^ 2w2

In the same manner all the following coefficients may be deter-

mined. It is easy to see that the definite integral 2 Jx sin nx F (x) dx

always has a determinate value, whatever the arbitrary functionF (x) may be. If the function F (x) be represented by thevariable ordinate of a line traced in any manner, the functionx F{x) sin nx corresponds to the ordinate of a second line whichcan easily be constructed by means of the first. The area boundedby the latter line between the abscissae x = 0 and x = X determinesthe coefficient a., i being the index of the order of the root n.

The arbitrary function F(x) enters each coefficient under thesign of integration, and gives to the value of v all the generalitywhich the problem requires; thus we arrive at the followingequation

sin nxx \ x sin n,x F (x) dx^L L P-kmH

sin n2x \x sin n2x F (x) dxe-

sin 2 2 ()&c.

This is the form which must be given to the general integralof the equation

<fo_h(<j?v 2dvdt~ \cLc* + x

in order that it may represent the movement of heat in a solidsphere. In fact, §11 the conditions of the problem are obeyed.

SECT. I.] ULTIMATE LAW OF TEMPERATURE. 2 7 7

1st, The partial differential equation is satisfied; 2nd, the quantityof heat which escapes at the surface accords at the same time withthe mutual action of the last layers and with the action of the air

on the surface; that is to say, the equation -r- + hx = 0, which

each part of the value of v satisfies when x = X, holds also whenwe take for v the sum of all these parts; 3rd, the given solutionagrees with the initial state when we suppose the time nothing.

292. The roots P19 n2, r?8, &c. of the equation

nX

are very unequal; whence we conclude that if the value of thetime is considerable, each term of the value of v is very small,relatively to that which precedes it. As the time of coolingincreases, the latter parts of the value of v cease to have anysensible influence; and those partial and elementary states, whichat first compose the general movement, in order that the initialstate may be represented by them, disappear almost entirely, oneonly excepted. In the ultimate state the temperatures of thedifferent layers decrease from the centre to the surface in thesame manner as in a circle the ratios of the sine to the arcdecrease as the arc increases. This law governs naturally thedistribution of heat in a solid sphere. When' it begins to exist,it exists through the whole duration of the cooling. Whateverthe function F (x) may be which represents the initial state, thelaw in question tends continually to be established ; and when thecooling has lasted some time, we may without sensible errorsuppose it to exist.

293. We shall apply the general solution to the case inwhich the sphere, having been for a long time immersed in afluid, has acquired at all its points the same temperature. Inthis case the function F(x) is 1, and the determination of thecoefficients is reduced to integrating x sin nx dx, from x = 0 tox = X: the integral is

sin nX — nX cos nX


Hence the value of each coefficient is expressed thus :

_ 2 sinrcX — nXcosnX %

n nX — sin nX cos nX'

the order of the coefficient is determined by that of the root n,the equation which gives the values of n being

nX cos nX ^ 7 ^-—: r r - = I — IIJL.

sin ?iX

We therefore find2 hXn nX cosec n Z — cos nX'

I t is easy now to form the general value which is given by theequation

vx _ e"fc^ sin rc^ 6 ~ ^ sin n2x „"T" 7 T^P

zf/: TTT I OCC.X— cos^ X)

Denoting by ev e2> 68, &c. the roots of the equation

tan e

and supposing them arranged in order beginning with the least;replacing nxX} n2X, n^X, &c. by et> e2, e3, &c, and writing instead

of k and h their values -^=r and -^, we have for the expression of

the variations of temperature during the cooling of a solid sphere,which was once uniformly heated, the equation

" = Z Ae ^ e, cosec et — cos et

X

€2^ e2 cosec e2 — cos e,X

+ &c. k

Ndte. tfh£ problem of the sphere has been very completely discussed byfeiemann, Partielle Differentialgleichungen, §§ 61—69. [A. F.]

SECT. II.] DIFFERENT REMARKS ON THIS SOLUTION. 279

SECTION II.

Different remarks on this solution.

294. We will now explain some of the results which may bederived from the foregoing solution. If we suppose the coefficienth, which measures the facility with which heat passes into the air,to have a very small value, or that the radius X of the sphere isvery small, the least value of e becomes very small; so that the

equation = 1 — -r-r X is reduced totan e K

h _ _ 6 )= 1 — ^ F >

n i X7-

or, omitting the higher powers of e, e 2 = - ^ - . On the other

hand, the quantity cos e becomes, on the same hypothesis,sin €

. ex2hX S m X—7^-. And the term — — is reduced to 1. On making theseK ex °

X

substitutions in the general equation we have v = e CJJX -f &c.We may remark that the succeeding terms decrease very rapidlyin comparison with the first, since the second root n2 is very muchgreater than 0; so that if either of the quantities h or X hasa small value, we may take, as the expression of the variations

of temperature, the equation v = € CJJX. Thus the different

spherical envelopes of which the solid is composed retain a

common temperature during the whole of the cooling. The

temperature diminishes as fche-ordinate of a logarithmic curve, the

time being taken for abscissa ; the initial temperature 1 is re-

duced after the time t to e CDX. In order that the initial

temperature may be reduced to the fraction —, the value of t

must be ^7 CD log m. Thus in spheres of the same material but


of different diameters, the times occupied in losing half or thesame defined part of their actual heat, when the exterior con-ducibility is very small, are proportional to their diameters. Thesame is the case with solid spheres whose radius is very small;and we should also find the same result on attributing to theinterior conducibility Z a very great value. The statement holds

7 XT

generally when the quantity - ^ is very small. We may regard

the quantity -^ as very small when the body which is being

cooled is formed of a liquid continually agitated, and enclosed ina spherical vessel of small thickness. The hypothesis is in somemeasure the same as that of perfect conducibility; the tem-perature decreases then according to the law expressed by the

Sht

equation v = e C1JA.

295. By the preceding remarks we see that in a solid spherewhich has been cooling for a long time, the temperature de-creases from the centre to the surface as the quotient of the sineby the arc decreases from the origin where it is 1 to the endof a given arc 6, the radius of each layer being representedby the variable length of that arc. If the sphere has a smalldiameter, or if its interior conducibility is very much greaterthan the exterior conducibility, the temperatures of the successivelayers differ very little from each other, since the whole arc ewhich represents the radius X of the sphere is of small length.The variation of the temperature v common to all its points

Sht

is then given by the equation v = e VDX. Thus, on comparing therespective times which two small spheres occupy in losing halfor any aliquot part of their actual heat, we find those timesto be proportional to the diameters.

Sht

296. The result expressed by the equation v = e CDX belongsonly to masses of similar form and small dimension. It has beenknown for a long time by physicists, and it offers itself as it werespontaneously. In fact, if any body is sufficiently small for thetemperatures at its different points to be regarded as equal, itis easy to ascertain the law of cooling. Let 1 be the initial

SECT. II.] EXTERIOR CONDUCIBILITIES COMPARED. 281

temperature common to all points; it is evident that the quantityof heat which flows during the instant dt into the mediumsupposed to be maintained at temperature 0 is hSvdt, denotingby 8 the external surface of the body. On the other hand,if G is the heat required to raise unit of weight from the tem-perature 0 to the temperature 1, we shall have CDV for theexpression of the quantity of heat which the volume V of thebody whose density is D would take from temperature 0 to

temperature 1. Hence jTjjy is the quantity by which the

temperature v is diminished when the body loses a quantity ofheat equal to hSvdt We ought therefore to have the equation

dv = njyp- > o r v = e •

If the form of the body is a sphere whose radius is X, we shall-2ht

have the equation v = eCDX.

297. Assuming that we observe during the cooling of thebody- in question two temperatures vx and v2 corresponding tothe times tx and £2, we have

hS _ log vt — log v2

CDV~ t ^ , -

We can then easily ascertain by experiment the exponent

If the same observation be made on different bodies, and ifwe know in advance the ratio of their specific heats G and C,we can find that of their exterior conducibilities h and h'.Keciprocally, if we have reason to regard as equal the valuesh and hf of the exterior conducibilities of two different bodies,we can ascertain the ratio of their specific heats. We see bythis that, by observing the times of cooling for different liquidsand other substances enclosed successively in the same vesselwhose thickness is small, we can determine exactly the specificheats of those substances.

We may further remark that the coefficient K which measuresthe interior conducibility does not enter into the equation

-Sht


Thus the time of cooling in bodies of small dimension does notdepend on the interior conducibility; and the observation of thesetimes can teach us nothing about the latter property; but itcould be determined by measuring the times of cooling in vesselsof different thicknesses.

298. What we have said above on the cooling of a sphereof small dimension, applies to the movement of heat in a thermo-meter surrounded by air or fluid. We shall add the followingremarks on the use of these instruments.

Suppose a mercurial thermometer to be dipped into a vesselfilled with hot water, and that the vessel is being cooled-freelyin air at constant temperature. It is required to find the lawof the successive falls of temperature of the thermometer.

If the temperature of the fluid were constant, and the thermo-meter dipped in it, its temperature would change, approachingvery quickly that of the fluid. Let v be the variable temperatureindicated by the thermometer, that is to say, its elevation abovethe temperature of the air; let u be the elevation of temperatureof the fluid above that of the air, and t the time correspondingto these two values v and u. At the beginning of the instantdt which is about to elapse, the difference of the temperatureof the thermometer from that of the fluid being v — u, the variablev tends to diminish and will lose in the instant dt a quantityproportional to v — u; so that we have the equation

dv = — h (v — u) dt

During the same instant dt the variable u tends to diminish,and it loses a quantity proportional to u, so that we have theequation

da = — Iludt.

The coefficient H expresses the velocity of the cooling of theliquid in air, a quantity which may easily be discovered by ex-periment, and the coefficient h expresses the velocity with whichthe thermometer cools in the liquid. The latter velocity is verymuch greater than H. Similarly we may from experimentfind the coefficient h by making the thermometer cool in fluidmaintained at a constant temperature. The two equations

du = — Hudt and dv = — It (v — it) dt,

SECT. II.] ERROR OF A THERMOMETER. 283

dvor u = Ae~m and -j- = - hv + hAe~m

dtlead to the equation

a and b being arbitrary constants. Suppose now the initial valueof v — u to be A, that is, that the height of the thermometerexceeds by A the true temperature of the fluid at the beginningof the immersion; and that the initial value of u is E. We candetermine a and b} and we shall have

The quantity v — u is the error of the thermometer, that isto say, the difference which is found between the temperatureindicated by the thermometer and the real temperature of thefluid at the same instant. This difference is variable, and thepreceding equation informs us according to what law it tendsto decrease. We see by the expression for the difference v — uthat two of its terms containing e~ht diminish very rapidly, withthe velocity which would be observed in the thermometer if itwere dipped into fluid at constant temperature. With respectto the term which contains e~m, its decrease is much slower,and is effected with the velocity of cooling of the vessel in air.I t follows from this, that after a time of no great length theerror of the thermometer is represented by the single term

HE _m Hrerm or

299. Consider now what experiment teaches as to the valuesof H and h. Into water at 8*5° (octogesimal scale) we dippeda thermometer which had first been heated, and it descendedin the water from 40 to 20 degrees in six seconds. This ex-periment was repeated carefully several times. From this wefind that the value of e~h is Q'0000421; if the time is reckonedin minutes, that is to say, if the height of the thermometer beE at the beginning of a minute, it will be E (0*000042) at theend of the minute. Thus we find

h log10 e = 4-3761271.

i 0*00004206, strictly. [A. F.]

284 THEORY OF HEAT. [CHAP. V;

At the same time a vessel of porcelain filled with water heatedto 60° was allowed to cool in air at 12°. The value of e~H inthis case was found to be 0*98514, hence that of .firiog10e is0*006500. We see by this how small the value of the fractione~h is, and that after a single minute each term multiplied bye~ht is not half the ten-thousandth part of what it was at thebeginning of the minute. We need not therefore take accountof those terms in the value of v - u. The equation becomes

Hu Hu H Hu

From the values found for H and h, we see that the latterquantity h is more than 673 times greater than H, that is tosay, the thermometer cools in air more than 600 times faster

than the vessel cools in air. Thus the term -y— is certainly less

than the 600th part of the elevation of temperature of the water

above that of the air, and as the term = ^ -y— is less thanft — n. ii

the 600th part of the preceding term, which is already very small,it follows that the equation which we may employ to representvery exactly the error of the thermometer is

Huv — u = -T— .

h

In general if H is a quantity very great relatively to h, wehave always the equation

HuV — U = - 7 - .

h

300. The investigation which we have just made furnishesvery useful results for the comparison of thermometers.

The temperature marked by a thermometer dipped into afluid which is cooling is always a little greater than that of thefluid. This excess or error of the thermometer differs with theheight of the thermometer. The amount of the correction willbe found by multiplying u the actual height of the thermometerby the ratio of H, the velocity of cooling of the vessel in air,to h the velocity of cooling of the thermometer in the fluid. Wemight suppose that the thermometer, when it was dipped into

SECT. II.] COMPARISON OF THERMOMETERS. 285

the fluid, marked a lower temperature. This is what almostalways happens, but this state cannot last, the thermometerbegins to approach to the temperature of the fluid; at the sametime the fluid cools, so that the thermometer passes first to thesame temperature as the fluid, and it then indicates a tempera-ture very slightly different but always higher.

300*. We see by these results that if we dip different thermo-meters into the same vessel filled with fluid which is coolingslowly, they must all indicate very nearly the same temperatureat the same instant. Calling h, h', h", the velocities of coolingof the thermometers in the fluid, we shall have

Hu Hu Huh h' ' h" '

as their respective errors. If two thermometers are equallysensitive, that is to say if the quantities h and h' are the same,their temperatures will differ equally from those of the fluid.The values of the coefficients h, h\ h" are very great, so that theerrors of the thermometers are extremely small and often in-appreciable quantities. We conclude from this that if a thermo-meter is constructed with care and can be regarded as exact, itwill be easy to construct several other thermometers of equalexactness. It will be sufficient to place all the thermometerswhich we wish to graduate in a vessel filled with a fluid whichcools slowly, and to place in it at the same time the thermometerwhich ought to serve as a model; we shall only have to observeall from degree to degree, or at greater intervals, and we mustmark the points where the mercury is found at the same timein the • different thermometers. These points will be at thedivisions required. We have applied this process to the con-struction of the thermometers employed in our experiments,so that these instruments coincide always in similar circum-stances.

This comparison of thermometers during the time of coolingnot only establishes a perfect coincidence among them, and rendersthem all similar to a single model; but from it we derive also themeans of exactly dividing the tube of the principal thermometer,by which all the others ought to be regulated. In this way we


satisfy the fundamental condition of the instrument, which is, thatany two intervals on the scale which include the same number ofdegrees should contain the same quantity of mercury. For therest we omit here several details which do not directly belong to-the object of our work.

301. We have determined in the preceding articles the tem-perature v received after the lapse of a time t by an interiorspherical layer at a distance x from the centre. It is requirednow to calculate the value of the mean temperature of the sphere,or that which the solid would have if the whole quantity of heatwhich it contains were equally distributed throughout the whole

mass. The volume of a sphere whose radius is x being —- —,

the quantity of heat contained in a spherical envelope whose

temperature is v, and radius x, will be vd[—— }. Hence the

mean temperature is

f4tTTX*\

o r

the integral being taken from x — 0 to x = X. Substitute for vits value

J e-kmH s i n n x + _2

x L x A x

and we shall have the equation

3 f 2 7 3 f sinn X-n.X cosn.X . 2

-ys p vax = -— \a^ l- h, ^~ e-Jcn&

We found formerly (Art. 293)

2 sin iiiX- n.XcoH nXas = — — * • i

SECT. II.] RADIUS OF SPHERE VERY GREAT. 287

We have, therefore, if we denote the mean temperature by z,

s^) j^f. ( [ s v - € 2 cos e / -'cub2J + 3 (2 i 2j3 . 4 ~~ e,8 (2€l - sin 2eJ + e2

3 (2e2 - sin

an equation in which the coefficients of the exponentials are allpositive.

302. Let us consider the case in which; all other conditionsremaining the same, the value X of the radius of the spherebecomes infinitely great1. Taking up the construction described

in Art. 285, we see that since the quantity -= r- becomes infinite,

the straight line drawn through the origin cutting the differentbranches of the curve coincides with the axis of x. We find thenfor the different values of e the quantities IT, 2-7r, 37r, etc.

__ _ AtSince the term in the value of z which contains e CI) X2

becomes, as the time increases, very much greater than thefollowing terms, the value of z after a certain time is expressed

Kn*without sensible error by the first term only. The index --^--

KIT2

being equal to ^ 2 , we see that the final cooling is very slowin spheres of great diameter, and that the index of e whichmeasures the velocity of cooling is inversely as the square of thediameter.

303. From the foregoing remarks we can form an exact ideaof the variations to which the temperatures are subject during thecooling of a solid sphere. The initial values of the temperatureschange successively as the heat is dissipated through the surface.If the temperatures of the different layers are at first equal, orif they diminish from the surface to the centre, they do notmaintain their first ratios, and in all cases the system tends moreand more towards a lasting state, which after no long delay issensibly attained. In this final state the temperatures decrease

1 Riemann has shewn, Part. Diff. gleich. § 69, that in the case of a very largesphere, uniformly heated initially, the surface temperature varies ultimately as thesquare root of the time inversely. [A. F.]


from the centre to the surface. If we represent the whole radiusof the sphere by a certain arc e less than a quarter of thecircumference, and, after dividing this arc into equal parts, takefor each point the quotient of the sine by the arc, this system ofratios will represent that which is of itself set up among thetemperatures of layers of equal thickness. From the time whenthese ultimate ratios occur they continue to exist throughout thewhole of the cooling. Each of the temperatures then diminishesas the ordinate of a logarithmic curve, the time being taken forabscissa. We can ascertain that this law is established by" ob-serving several successive values z> z', z'\ / " , etc., which denotethe mean temperature for the times t, t + ®, t + 20, t + 3®, etc.;the series of these values converges always towards a geometrical

z z z"progression, and when the successive quotients - , — , —,, etc.

z z zno longer change, we conclude that the relations in question areestablished between the temperatures. When the diameter of thesphere is small, these quotients become sensibly equal as soon asthe body begins to cool. The duration of the cooling for a giveninterval, that is to say the time required for the mean tem-perature z to be reduced to a definite part of itself — , increasesas the diameter of the sphere is enlarged.

304. If two spheres of the same material and differentdimensions have arrived at the final state in which whilst thetemperatures are lowered their ratios are preserved, and if wewish to compare the durations of the same degree of cooling inboth, that is to say, the time © which the mean temperature

of the first occupies in being reduced to —, and the time © inm

which the temperature z of the second becomes —, we mustm

consider three different cases. If the diameter of each sphere issmall, the durations © and ©' are in the same ratio as thediameters. If the diameter of each sphere is very great, thedurations © and ©' are in the ratio of the squares of thediameters; and if the diameters of the spheres are includedbetween these two limits, the ratios of the times will be greaterthan that of the diameters, and less than that of their squares.

SECT. II.] EQUATION OF CONDITION HAS ONLY REAL ROOTS. 289

The exact value of the ratio has been already determined1.The problem of the movement of heat in a sphere includes thatof the terrestrial temperatures. In order to treat of this problemat greater length, we have made it the object of a separatechapter2.

305. The use which has been made above of the equation

- = X is founded on a geometrical construction which is very-tan 6well adapted to explain the nature of these equations. The con-struction indeed shows clearly that all the roots are real; at thesame time it ascertains their limits, and indicates methods fordetermining the numerical value of each root. The analyticalinvestigation of equations of this kind would give the same results.First, we might ascertain that the equation e — X tan e = 0, inwhich X is a known number less than unity, has no imaginaryroot of the form m + nj— 1. It is sufficient to substitute thisquantity for e; and we see after the transformations that the firstmember cannot vanish when we give to m and n real values,unless n is nothing. It may be proved moreover that there canbe no imaginary root of any form whatever in the equation

€ cos 6 — X sin €6 — X tan e = 0, or = 0.

cose

In fact, 1st, the imaginary roots of the factor = 0 do not' & J cose

belong to the equation e — X tan e = 0, since these roots are all of

the form m + nj— 1; 2nd, the equation sin € — - cos e = 0 hasX

necessarily all its roots real when X is less than unity. To provethis proposition we must consider sin e as the product of theinfinite number of factors

1 It is O : Q^e^X2 : e^Y'2, as may be inferred from the exponent of the firstterm in the expression for z, Art. 301. [A. F.]

2 The chapter referred to is not in this work. It forms part of the Suite dumemorie sur la theorie du mouvement de la chaleur dans les corps solides. See note,page 10.

The first memoir, entitled Theorie du mouvement de la chaleur dans les corpssolides, is that which formed the basis of the Theorie analytique du mouvement dela chaleur published in 1822, but was considerably altered and enlarged in thatwork now translated. [A. F.]

F. II. 19

290 THEORY OF HEAT. [CHAP. V,

and consider cos e as derived from sin e by differentiation.

Suppose that instead of forming sin e from the product of aninfinite number of factors, we employ only the m first, and denotethe product by <£m(e). To find the corresponding value of cose,we take

This done, we have the equation

Now, giving to the number m its successive values 1, 2, 3, 4, &c.from 1 to infinity, we ascertain by the ordinary principles ofAlgebra, the nature of the functions of e which correspond tothese different values of m. We see that, whatever m the numberof factors may be, the equations in e which proceed from themhave the distinctive character of equations all of whose rootsare real. Hence we conclude rigorously that the equation

tan e '

in which X is less than unity, cannot have an imaginary root1.The same proposition could also be deduced by a different analysiswhich we shall employ in one of the following chapters.

Moreover the solution we have given is not founded on theproperty which the equation possesses of having all its rootsreal. It would not therefore have been necessary to provethis proposition by the principles of algebraical analysis. Itis sufficient for the accuracy of the solution that the integralcan be made to coincide with any initial state whatever; forit follows rigorously that it must then also represent all thesubsequent states.

1 The proof given by Eiemann, Part. Biff. Gleich. § 67, is more simple. Themethod of proof is in part claimed by Poisson, Bulletin de la Societe Philomatique,PariB, 1826, p. 147. [A. F.].

CHAPTER VI.

OF THE MOVEMENT OF HEAT IN A SOLID CYLINDER.

306. THE movement of heat in a solid cylinder of infinitelength, is represented by the equations

dv K (d2v ldv\ - hfo CJ9 d* dj K dx

which we have stated in Articles 118, 119, and 120. To inte-grate these equations we give to v the simple particular valueexpressed by the equation v = ue~mi; m being any number, and

u a function of x. We denote by k the coefficient ^ which

enters the first equation, aiid by h the coefficient -^ which enters

the second equation. Substituting the value assigned to v, wefind the following condition

m d\ 1 du ^r t t + -7-2 + - IT" = 0.A; ^

Next we choose for u a function of x which satisfies thisdifferential equation. It is easy to see that the function maybe expressed by the following series

• , - 1 9X* , # V A ° , &cM " l"" 22 22 . 4a 22 .42 . 62 '

5f denoting the constant -j-. We shall examine more particularly

in the sequel the differential equation from which this series

19—2

292 THEORY OF HEAT. [CHAP. VI.

is derived; here we consider the function u to be known, andwe have ue~0kt as the particular value of v.

The state of the convex surface of the cylinder is subjectto a condition expressed by the definite equation

which must be satisfied when the radius x has its total value X;whence we obtain the definite equation

h / X2 g*X* g*X° „ \\ 9 ~¥ + 2\T2 ~ W7i\ 62 7

2 2 4 2 6 2" 22 2 * . 42 2 2 . 4 2 . 6

thus the number g which enters into the particular value ue~°u

is not arbitrary. The number must necessarily satisfy thepreceding equation, which contains g and X

We shall prove that this equation in g in which h and Xare given quantities has an infinite number of roots, and thatall these roots are real. It follows that we can give to thevariable v an infinity of particular values of the form ue~oU,which differ only by the exponent g. We can then composea more general value, by adding all these particular valuesmultiplied by arbitrary coefficients. This integral which servesto resolve the proposed equation in all its extent is given bythe following equation

v = ajjixe-^u 4- a2u2e-9*H + a^e"^ + &c,

9\9 9*> 9s> &°- -denote all the values of g which satisfy the definiteequation; uv u^ uv &c. denote the values of u which correspondto these different roots; ax, a2, a3, &c. are arbitrary coeffi-cients which can only be determined by the initial state of thesolid.

307. We must now examine the nature of the definiteequation which gives the values of g, and prove that all the rootsof this equation are real, an investigation which requires attentiveexamination.

CHAP. VI.] THE EQUATION OF CONDITION. 293

In the series

which expresses the value which u receives when x = X, we shall

replace ^ - by the quantity 0, and denoting this function of 0

by / (6) or 2/, we have/)2 /]3 /J4(7 1/ C7

the definite equation becomes

7-, + &c.2s + 3 2 2 . 32 "" 4 2 2 .

"

denoting the function ^ ^

Each value of 9 furnishes a value for g} by means of theequation

and we thus obtain the quantities gv g2, g3) &c. which enter ininfinite number into the solution required.

The problem is then to prove that the equation

2 +

must have all its roots real. We shall prove in fact that theequation f{&) — 0 has all its roots real, that the same is thecase consequently with the equation f\6) =0 , and that it followsthat the equation

A -

has also all its roots real, A representing the known number

hX

2 9 4 THEORY OF HEAT. [CHAP. VI.

308. The equation

on being differentiated twice, gives the following relation

We write, as follows, this equation and all those which maybe derived from it by differentiation,

&c,

and in general

Now if we write in the following order the algebraic equationX = 0, and all those which may be derived from it by differentiation,

X = 0 , -=- = 0, 1-5-= 0, -7-3 = 0, &c,

and if we suppose that every real root of any one of these equa-tions on being substituted in that which precedes and in that whichfollows it gives two results of opposite sign; it is certain that theproposed equation X = 0 has all its roots real, and that conse-quently the same is the case in all the subordinate equations

These propositions are founded on the theory of algebraic equa-tions, and have been proved long since. It is sufficient to provethat the equations

fulfil the preceding condition. Now this follows from the generalequation

CHAP. VI.] REALITY OF THE ROOTS. 295

for if we give to 0 a positive value which makes the fluxion ,^.^

dly di+1yvanish, the other two terms -^ and ^.^ receive values of opposite

sign. With respect to the negative values of 0 it is'evident, fromthe nature of the function / (#) , that no negative value substitutedfor 0 can reduce to nothing, either that function, or any of theothers which are derived from it by differentiation: for the sub-stitution of any negative quantity gives the same sign to all theterms. Hence we are assured that the equation y = 0 has all itsroots real and positive.

309. It follows from this that the equation f (0) = 0 or y} = 0

also has all its roots real; which is a known consequence from the

principles of algebra. Let us examine now what are the suc-

cessive values which the term 0 ~F77y: or 0 — receives when we give

to 0 values which continually increase from 0 = 0 to 0 = GO . If a

value of 0 makes y nothing, the quantity 0 — becomes nothing

also ; it becomes infinite when 0 makes y nothing. Now itfollows from the theory of equations that in the case in question,every root of y = 0 lies between two consecutive roots of y = 0,and reciprocally. Hence denoting by 0X and 03 two consecu-tive roots of the equation y = 0, and by 02 that root of theequation y = 0 which lies between 0X and 03, every value of 0 in-cluded between 0t and 02 gives to y a sign different from thatwhich the function y would receive if 0 had a value included be-

tween 02 and 03. Thus the quantity 0 — is nothing when 0 = 0X\ it

is infinite when 0 = 02, and nothing when 0 = 03. The quantity 0 —

must therefore necessarily take all possible values, from 0 to in-finity, in the interval from 0 to 02, and must also take all possiblevalues of the opposite sign, from infinity to zero, in the interval

from 02 to 03. Hence the equation A = 0— necessarily has one

real root between Qx and 6S and since the equation y = 0 has all its

roots real in infinite number, it follows that the equation A = 6^-

has the same property. In this manner we have achieved theproof that the definite equation

hY 2* - 2 ' 4 " - 2 * 4"6"'~&C<

1 *L 4- Y- *L—' u RLQ.

22 2a 42 22 42 g2

in which the unknown is g, has all its roots real and positive. Weproceed to continue the investigation of the function it and of thedifferential equation which it satisfies.

flu ill 1310. From the equation y-^-jn + O ~ijL — ®>we derive the general

dly di+1y di+*yequation -^. + (i + 1) ^ + 0 j^i+i ~ 0, and if we suppose 0 = 0 wehave the equation

di+ly _ 1 d'y

which serves to determine the coefficients of the different terms ofthe development of the function/(0), since these coefficients dependon the values which the differential coefficients receive when thevariable in them is made to vanish. Supposing the first term tobe known and to be equal to 1, we have the series

y = l - 0 + — - - _ + 2<2 g2 42 - &c.

If now in the equation proposed

^ 1 & _dx2 x dx ~~

x2

we make g^ = 0, and seek for the new equation in u and 6, re-

garding u as a function of 0, we find

CHAP. VI.] SUM OF A CERTAIN SERIES. 297

Whence we conclude

or u - i - 2 , + 22 42 2 , 4>2< 6 , + occ.

It is easy to express the sum of this series. To obtain theresult, develope as follows the function cos (a sin x) in cosines ofmultiple arcs. We have by known transformations

2 cos (a sin x) =e^~l e-i

ae"xy/~l + ^ l ^ ' 1

ei**-*y/-1 y

and denoting ex ~1 by to,aw a<o~* aw aw~^

2 cos (a sin x) = e*e""2" + e~Te^~.

Developing the second member according to powers of co, wefind the term which does not contain co in the development of2 cos (a sin x) to be

22 2 2 .4 2 2 2 .4 2 .6 2

The coefficients of co1, cos, co5, &c. are nothing, the same is the casewith the coefficients of the terms which contain co'1, co~s, co~5, &c.;the coefficient of co~2 is the same as that of co2; the coefficient of co4 is

a6

+ 7 ;4.6 .8 2 2 . 4 . 6 . 8 . 1 0

the coefficient of to"4 is the same as that of to4. It is easy to expressthe law according to which the coefficients succeed j but withoutstating it, let us write 2 cos 2x instead of (to2 + co"2), or 2 cos 4# in-stead of (to4 + to"4), and so on : hence the quantity 2 cos (a sin x) iseasily developed in a series of the form

A + B cos 2x + C cos 4# + D cos 6# + &c,

and the first coefficient A is equal to~2

2(i-|2 + ^ -~ ;+&c. ) ;

if we now compare the general equation which we gave formerly

1 If f•^7rcp(x) = 2 j(f)(x)dx +cos x j(j)(


with the equation

2 cos (a sin x) = A + B cos 2x + G cos 4tx + &c,

we shall find the values of the coefficients A, B, C expressed bydefinite integrals. It is sufficient here to find that of the firstcoefficient A. We have then

- A = — (cos (a sin x) dx,Z 7TJ

the integral should be taken from x — 0 to x — IT. Hence the2 a * 6

value of the series 1 — ^ + ^—j:2 — -2—-^—-^ + &c. is that of the

definite integral dx cos (a sin x). We should find in the sameJo

manner by comparison of two equations the values of the successivecoefficients B, C, &c; we have indicated these results because theyare useful in other researches which depend on the same theory.It follows from this that the particular value of u which satisfiesthe equation

d2u ldu . 1+ + 0

the integral being taken from r = 0 to r = IT. Denoting by q this

value of u3 and making u = qS, we find 8 = a + b I—% , and we haveJxq2'

as the complete integral of the equation qu + ~ + -C^ = 0* dx x dr

u=\a + b I —r? - p /cos (x Jg sin r) dr.L J ^ jjcos (xjg sin r) c/rf J J

a and 6 are arbitrary constants. If we suppose b = 0, we have,as formerly,

u = I cos (a? sin r) dr%

With respect to this expression we add the following remarks.

311. The equation

os (0sin u) dn = 1 - | + ^L - ^L^ + &C.

CHAP. VI.] VERIFICATION OF THE SUM. 299

verifies itself. We have in fact

f t* • w [j A. 02sin2w , ^s in 4 ^ 6>6sin6a p \Jcos (0 sin u)du= Idull ^ 1 ,, n; h &c. I;

and integrating from u = 0 to w = 7r, denoting by $,, #4, Sa, &c.the definite integrals

we have

I sin2 u du, I sin4 u du, I sin6 u du, &c,

f 02 #4 0°I COS (6 Sin tt) Ji6 = 7T — TX- ^ 2 4- jr S4 — |pr ^S6 + &C,J c. li p.it remains to determine >S2, ^4, ^6, &c. The term sinn u, n being

an even number, may be developed thus

sin" u = AnJt Bn cos 2u + Cn cos 4<u -f &c.

Multiplying by du and integrating between the limits u = 0 and

^ = 7T, we have simply /sin t tMd^= ^4n7r, the other terms vanish.

Prom lha known formula for the development of the integralpowers of sines, we have

A - L 2 , 1 M . _ 1 _ 4 .5 ,6^ 2 - 2 ? • i > ^ 2

4 ' 1 . 2 ' 6 " " 2 6 # 1 . 2 . 3 '

Substituting these values of S2> S4, S# &c, we find

- J cos (0 sin M) rfM= 1 _ ~ + ^—^ - 2. 4. fi2 + &c.

We can make this result more general by taking, instead ofcos (t sin it), any function whatever cf> of t sin u.

Suppose then that we have a function {t sin u) = (f> + tfy' sin u + TX >" sin2 w + -jo </>"' si^ +

and^fdu4>(f sin u) = <!> + t S^'

Now, it is easy to see that the values of 8V S9, 8& &c. arenothing. With respect to 82, 8# 8a, &c. their values are thequantities which we previously denoted by A2, A^ A6, &c. Forthis reason, substituting these values in the equation (e) we havegenerally, whatever the function + ? <j>" + ^ & + Y - £ r j t F + '&c.,

in the case in question, the function <fr (z) represents cos z, and wehave 0 = 1, <£" = - 1, ftv = 1, </>* = - 1, and so on.

312. To ascertain completely the nature of the function/(0),and of the equation which gives the values of g, it would benecessary to consider the form of the line whose equation is

m AS

l 0 + + &

which forms with the axis of abscissae areas alternately positiveand negative which cancel each other; the preceding remarks, also,on the expression of the values of series by means of definiteintegrals, might be made more general. When a function of thevariable x is developed according to powers of x, it is easy todeduce the function which would represent the same series, if thepowers x, x2, xB, &c. were replaced by cos x, cos 2x9 cos 3x, &c. Bymaking use of this reduction and of the process employed in thesecond paragraph of Article 235, we obtain the definite integralswhich are equivalent to given series; but we could not enter uponthis investigation, without departing too far from our main object..

It is sufficient to have indicated the methods which haveenabled us to express the values of series by definite integrals.

We will add only the development of the quantity 6 TTW in a

continued fraction.

313. The undetermined y orf(0) satisfies the equation

CHAP. VI.] CORRESPONDING CONTINUED FRACTION. 301

whence we derive, denoting the functions

whence

&c;

we conclude

yy

- 11 -

dd'

' or

l..iv

e2 -

d'ffz

r

y~

y

y

e3 -

<£yy 1'/J4 )

-yy + ,

_ *3y H

04 - 5

&c.

* 1-

0— &c.

_ 1

f 7

+ ¥

f)~f'(ff\

Thus the value of the function — -^-/AT which enters into thedefinite equation, when expressed as an infinite continuedfraction, is

_i_ JL A A g1 - 2 - 3 - 4 - 5 - & c /

314. We shall now state the results at which we have up tothis point arrived.

If the variable radius of the cylindrical layer be denoted by x,and the temperature of the layer by v, a function of x and thetime t; the required function v must satisfy the partial differentialequation

dv __ ., /d2v 1 dv\dt~ KM* xTx) >

for v we may assume the following valuei) = ue~mt;

u is a function of x, which satisfies the equationm dht 1 du


If we make 0 = -=-—, and consider u as a function of x, we havefc 2t

du

The following value

satisfies the equation in u and 0. We therefore assume the valueof u in terms of x to be

- ma? m2 of — w* x3 „W" i 2 i + F22.T2 JF2".4".6"+ '

the sum of this series isI f / m . \ ,- cos [x A / i- smr\dr\irj \ V k )

the integral being taken from r = 0 to r = 7r. This Value of t? interms of # and m satisfies the differential equation, and retains a

finite value when x is nothing. Further, the equation hu + -y- = 0

must be satisfied when x — X the radius of the cylinder. Thiscondition would not hold if we assigned to the quantity m anyvalue whatever; we must necessarily have the equation

2 "" 1 _ 2 - 2T= 4 - 5 - &c.'

in which 0 denotes j - -^ -

This definite equation, which is equivalent to the following,

gives to 0 an infinity of real values denoted by 0V 02, 03, &c.; thecorresponding values of m are

1 2 3

thus a particular value of v is expressed by* WktOt r

TTV = c"1 X2 J xcos f 2^j01 sin

CHAP. VI.] FORM OF THE GENERAL SOLUTION. 303

We can write, instead of 0V one of the roots 0V 02, 03, &c, andcompose by means of them a more general value expressed bythe equation

a2e

Icosf 2-^ slQx sin q) dq

r /

I cos 12

r

ai> a2> av ^c- a r e arbitrary coefficients: the variable q dis-appears after the integrations, which should be taken from q — 0to q = 7T.

315. To prove that this value of v satisfies all the conditionsof the problem and contains the general solution, it remains onlyto determine the coefficients av a2, az> &c. from the initial state.Take the equation

v = axe~m^ ut + a^e~m^ u2 + aze~m^ uz + &c,

in which uv u2, us, &c. are the different values assumed by thefunction w, or

when, instead of -j-9 the values gv g2J g^ &c. are successively sub-

stituted. Making in it t = 0, we have the equation

F=* ajn% + a2u2 + a3uz + &c,

in which T i s a given function of x. Let <f> (a?) be this function;if we represent the function u% whose index is % by ^ (x Jg.), wehave

<f> (x) = a& (00 Jgx) + a2f (x sjg^ + a3f {x Jgz) + &c.

To determine the first coefficient, multiply each member ofthe equation by crx dx, <rt being a function of x, and integrate fromx = 0 to x = X. We then determine the function xrx, so that afterthe integrations the second member may reduce to the first termonly, and the coefficient ax may be found, all the other integrals

having nul values. Similarly to determine the second coefficienta2, we multiply both terms of the equation

<£ (<v) = a2ux + a2u2 + asua + &c.

by another factor <r2 dx, and integrate from x = 0 to x = X Thefactor <r2 must be such that all the integrals of the second membervanish, except one, namely that which is affected by the coefficienta2. In general, we employ a series of functions of x denoted by°~i> a2> G^ &c- w n i c n correspond to the functions ulf u2, u3, &c.;each of the factors a has the property of making all the termswhich contain definite integrals disappear in integration exceptone; in this manner we obtain the value of each of the coefficientsax, a2> a3, &c. We must now examine what functions enjoy theproperty in question.

316. Each of the terms of the second member of the equation

is a definite integral of the form alaudx; u being a function of x

which satisfies the equation

m d2u 1 duk ax x ax

we have therefore a andx = — a— -I - -y- + <r-r— .J in J \x ax ax J

Developing, by the method of integration by parts, the terms

du 7 1 [ d2u 7d d d

we have - -j-dx = C+u lud(-Jx ax x J \x)x ax

, f d?u , _. du da f d2a 7and a , 2 dx = D •+ -y-a —U--,- + \u-y~ dx.J ax dx dx J dx

The integrals must be taken between the limits x = 0 andx = X, by this condition we determine the quantities which enterinto the development, and are not under the integral signs. To in-dicate that we suppose x = 0 in any expression in x, we shall affectthat expression with the suffix a; and we shall give it the suffixco to indicate the value which the function of x takes, when wegive to the variable x its last value X

CHAP. VI.] AUXILIARY MULTIPLIERS. 305

Supposing x = 0 in the two preceding equations we have

xj

thus we determine the constants C and D. Making then x = X inthe same equations, and supposing the integral to be taken fromx = 0 to x = Xy we have

adu 7 / <r\ / <r\ f , f<r\- ^ - d x = i u - ) — [ w - J — I w a - J

and J - ^ ^ = - - ^ J w - ( ^ - - ^ j + J"^thus we obtain the equation

d(-)d2a- \xj) 7 (du da

+

- c du da a-j- a — u -j—\-u-dx ax xt

317. If the quantity -j-2 - ^ which multiplies u under the

sign of integration in the second member were equal to the pro-duct of o" by a constant coefficient, the terms

would be collected into one, and we should obtain for the required

integral jaudx a value which would contain only determined quan-

tities, with no sign of integration. It remains only to equate that

value to zero.

Suppose then the factor a to satisfy the differential equation of•j(o\

d2a Ix)the second order T a + -j^ r—• = 0 in the same manner as the

k dx axfunction u satisfies the equation

m d2u 1 du _k dx2 x dx y

F. H. 20

m and n being constant coefficients, we have

n — m[ , (du da a\ (du da o\k J \dx dx x/n \dx dx x/a

Between u and a a very simple relation exists, which is dis-

. . . ,, n d2a \xj ~covered when in the equation r d + T^ -^— = 0 we suppose

a = xs ; as the result of this substitution we have the equation

n d2s 1 ds _ 0

k dx2 x dx y

which shews that the function s depends on the function u givenby the equation

m d2u 1 du -T u + :r-2 + - T~ = °-tc ax x ax

To find s it is sufficient to change m into n in the value of. u;

the value of i rhas been denoted by ^ [x/u -r\, that of a will

therefore be ic-^ [x\I -=- j .

We have then

du dcr <r- r c — u-y- + u-dx dx x

the two last terms destroy each other, it follows that on makingx = 0, which corresponds to the suffix a, the second membervanishes completely. We conclude from this the following equa-tion

Y / m ; K f \ / 3 .._ _ /n= — X\/-1 T=- and hX=-X\/ T

n

CHAP. VI.] VANISHING FORM. 3 0 7

It is easy to see that the second member of this equation isalways nothing when the quantities m and n are selected fromthose which we formerly denoted by mv m2, m3, &c.

We have in fact

hX:

comparing the values of hX we see that the second member of theequation (f) vanishes.

It follows from this that after we have multiplied by adx thetwo terms o the equation

<f> (%) = axux + a2u2 + a3u3 + & c ,

and integrated each side from % = 0 to x = X, in order that each ofthe terms of the second member may vanish, it suffices to take

for a the quantity xu or a

We must except only the case in which n = m, when the value

of \audx derived from the equation ( / ) is reduced to the form - ,

and is determined by known rules.

318. If A/J = p and . A / | = v, we have

If the numerator and denominator of the second member areseparately differentiated with respect to p, the factor becomes, onmaking JJL = v,

We have on the other hand the equation

d2u \du rt

20—2

308

and also

or,

hence we have

THEORY OF HEAT. [CHAP. VI.

we can therefore eliminate the quantities y\r and ijr" from theintegral which is required to be evaluated, and we shall find as thevalue of the integral sought

putting for /i its value, and denoting by E£ the value which the

function u or ^ ( # / v / x ) âê s w n e n w e suppose x = X. The

index i denotes the order of the root m of the definite equa-tion which gives an infinity of values of m. If we substitute

mi or — ~ m [l-\ 1, we have

319. It follows from the foregoing analysis that we have thetwo equations

ocujUn dx = 0 and I xu% ax = \ 1 +1 — T = I V — ^ ,ô ( \2 sjQJ ) &

the,first holds whenever the number i and j are different, and thesecond when these numbers are equal.

Taking then the equation &c. are to be determined, we shallfind the coefficient denoted by ai by multiplying the two membersof the equation by xu^x, and integrating from x = 0 to x = X;the second member is reduced by this integration to one termonly, and we have the equation

2 jx<f> (x) utdx = a, X'US j l + ( ^ L J J ,

CHAP. VI.] COMPLETE SOLUTION. 309

which gives the value of a.. The coefficients av a2> a3,... aP beingthus determined, the condition relative to the initial state expressedby the equation <j> (x) = afix + a2u2 + aBu3 -f &c, is fulfilled.

We can now give the complete solution of the proposed problem;it is expressed by the following equation:

rx rx™ I x<f>(x)u1dx ^IMQ I x<j>(x)u2dx J^Q

= - 2 U £ X% +L~7 F T ^ T ^

+ &C.

The function of x denoted by u in the preceding equation isexpressed by

I cos f - ^ r ^ s i n q) dq;

all the integrals with respect to x must be taken from x = 0 tox = X, and to find the function u we must integrate from q = 0 toq = IT; <j) (x) is the initial value of the temperature, taken in theinterior of the cylinder at a distance x from the axis, whichfunction is arbitrary, and 6V 02, 03, &c. are the real and positiveroots of the equation

hX_ J_ JL A. 1_ °2" " 1 - 2 - 3 - 4 - 5-&c. '

320. If we suppose the cylinder to have been immersed foran infinite time in a liquid maintained at a constant temperature,the whole mass becomes equally heated, and the function <f> (x)which represents the initial state is represented by unity. Afterthis substitution, the general equation represents exactly thegradual progress of the cooling.

If t the time elapsed is infinite, the second member containsonly one term, namely, that which involves the least of all theroots 6V 62, 03, &c; for this reason, supposing the roots to bearranged according to their magnitude, and 6 to be the least, thefinal state of the solid is expressed by the equation

vX2


From the general solution we might deduce consequencessimilar to those offered by the movement of heat in a sphericalmass. We notice first that there are an infinite number ofparticular states, in each of which the ratios established betweenthe initial temperatures are preserved up to the end of the cooling.When the initial state does not coincide with one of these simplestates, it is always composed of several of them, and the ratios ofthe temperatures change continually, according as the time increases.In general the solid arrives very soon at the state in which thetemperatures of the different layers decrease continually preservingthe same ratios. When the radius X is very small1, we find that

2h

the temperatures decrease in proportion to the fraction e"cnxm

If on the contrary the radius X is very large2, the exponent ofe in the term which represents the final system of temperaturescontains the square of the whole radius. We see by this whatinfluence the dimension of the solid has upon the final velocity ofcooling. If the temperature8 of the cylinder whose radius is X,passes from the value A to the lesser value B, in the time T, thetemperature of a second cylinder of radius equal to X' will passfrom A to B in a different time T\ If the two sides are thin, theratio of the times T and T will be that of the diameters. If, onthe contrary, the diameters of the cylinders are very great, theratio of the times T and T will be that of the squares of thediameters.

1 When X is very small, $ = — , from the equation in Art. 314. Hencez

WMQ JM.te X2 becomes e x .

In the text, h is the surface conducibility.2 When X is very large, a value of 0 nearly equal to one of the roots of the

6 0 B ftquadratic equation 1= — — — - will make the continued fraction in Art. 314

assume its proper magnitude. Hence 0=1-446 nearly, and

e x* becomes e X2 .The least root of f(0) = O is 1-4467, neglecting terms after 04.

3 The temperature intended is the mean temperature, which is equal to

[A. F.]

CHAPTER VII.

PROPAGATION OF HEAT IN A RECTANGULAR PRISM.

^7^7) (1?J ft ti

321. THE equation T - 2 + J i + "n = 0; which we have stated1 ax ay azrin Chapter IT., Section IV., Article 125, expresses the uniform move-ment of heat in the interior of a prism of infinite length, sub-mitted at one end to a constant temperature, its initial tempera-tures being supposed nul. To integrate this equation we shall,in the first place, investigate a particular value of v, remarkingthat this function v must remain the same, when y changes signor when z changes sign; and that its value must become infinitelysmall, when the distance x is infinitely great. From this it iseasy to see that we can select as a particular value of v thefunction ae~mx cos ny cos pz ; and making the substitution we findm2 — n3 — p2 = 0. Substituting for n and p any quantities what-ever, we have m — Jtf+p2. The value of v must also satisfy the

definite equation J.V + ~J~ — Q> when y = l or — ly and the equation

T H T ^ = 0 when z = I or - 1 (Chapter II., Section IV., Article 125).

If we give to v the foregoing value, we have

— n sin ny + T COS ny = 0 and — p smpz + -r cospz = 0,

or -r = pi tan pi, ^r — ^l tan nl.

We see by this that if we find an arc e, such that e tan e is equal

to the whole known quantity y I, we can take for n or p the quan-

312 THEORY OF HEAT. [CHAP. VII.

tity y. Now, it is easy to see that there are an infinite number

of arcs which, multiplied respectively by their tangents, give the

same definite product -j-f whence it follows that we can find

for n or p an infinite number of different values.

322. If we denote by el9 e2, e3, &c. the infinite number of

arcs which satisfy the definite equation e tan e = -j-, we can take

for n any one of these arcs divided by I The same would be thecase with the quantity p\ we must then take m2 — n2 + p2. If wegave to n and p other values, we could satisfy the differentialequation, but not the condition relative to the surface. We canthen find in this manner an infinite number of particular valuesof v, and as the sum of any collection of these values still satisfiesthe equation, we can form a more general value of v.

Take successively for n and p all the possible values, namely,

t' if' 7*' &C' D e D o t i n § by av a*> a*> &a> h> K h> &c., con-stant coefficients, the value of v may be expressed by the followingequation:

2 cos nxy + a2e~xy/n*2+ni* cos n2y + &c.) bx cos nxz

+ {axe-x v^2+^2 cos nxy + a2e'x v ^ 2 + ^ 2 cos n2y + &c.) b2 cos n2z

4- (axe'x Vwi2+Wa2 cos njj + a#-**»**+»* cos n2y + &c.) 63 cos nzz

323. If we now suppose the distance x nothing, every point ofthe section A must preserve a constant temperature. It is there-fore necessary that, on making x = 0, the value of v should bealways the same, whatever value we may give to y or to z; pro-vided these values are included between 0 and I. Now, on makingx = 0, we find

v = {ax cos nxy + a2 cos n2y + az cos nzy + &c.)

x (bx cos nxz + b2 cos n2y + b3 cos n3y + &c).

CHAP. VII.] DETERMINATION OF THE COEFFICIENTS. S I 3

Denoting by 1 the constant temperature of the end A, assumethe two equations

1 = ax cos nxy + a2 cos n2y + a3 cos nsy + &c,

1 = bx cos nxy + b2 cos n2y + bs cos n3y + &c.

It is sufficient then to determine the coefficients alf a2, a3, &c,whose number is infinite, so that the second member of the equa-tion may be always equal to unity. This problem has alreadybeen solved in the case where the numbers n19 n2, n3, &c. form theseries of odd numbers (Chap. III., Sec. II., Art. 177). Herenx, n2, n3, &c. are incommensurable quantities given by an equa-tion of infinitely high degree.

324. Writing down the equation

1 = ax cos nxy + a2 cos n2y + as cos n3y + &c,

multiply the two members of the equation by cos nty dy, and takethe integral from y = 0 to y = I. We thus determine the firstcoefficient av The remaining coefficients may be determined in asimilar manner.

In general, if we multiply the two members of the equation bycos vy, and integrate it, we have corresponding to a single termof the second member, represented by a cos ny, the integral

a /cosny cos vydy or ^a \ cos (n — v)y dy +-^d /cos (n + v)ydy^

0T>

and making y = I,

a ((n + v) sin (n — v) I + (n — v) sin (n -f- v) I

Now, every value of n satisfies the equation ft tan ^ = -7; the

same is the case with v, we have therefore

n tan vl = v tan vl;

or n sin nl cos vl — v sin vl cos ??Z = 0.


Thus the foregoing integral, which reduces to

Ob V

-2 2 \n s i n n^ c o s vl — r c o s w ^ s i n ^0*

is nothing, except only in the case where n = v. Taking then theintegral

a (sin (n — v)l sin (n + v) I)

2} 7i — ^ n-\-v y

we see that if we have n = v> it is equal to the quantity

sin 27

It follows from this that if in the equation

1 = ax cos n^y + a2 cos n2y + a3 cos n3y + &c.

we wish to determine t t e coefficient of a term of the secondmember denoted by a cos ny, we must multiply the two membersby cos nydy, and integrate from y = 0 to y = L We have theresulting equation

, s i n 2nl

whence we deduce — ^—^—J = 7 ft. In this manner the coeffi-2nZ + sin 2nl 4

cients ax, a2, a3, &c. may be determined; the same is the casewith b19 b2y 63, &c, which are respectively the same as the former.coefficients.

325. It is easy now to form the general value of v. 1st, itd2v d2v d2v

satisfies the equation _ + _ + _ = (); 2nd, it satisfies the two

conditions k-j- + hv = 0, and k-^ + hv = 0; 3rd, it gives a constant

value to v when we make x = 0, whatever else the values of y andz may be, included between 0 and l\ hence it is the completesolution of the proposed problem.

We have thus arrived at the equation

1 _ sin ntl cos nxy sin n} cos n2y sin n3l cos njj4 2l i l + J + ^4 2nxl + sin 2 '

CHAP. VII.] THE SOLUTION. 315

or denoting by el9 e2, e3, &c. the arcs nj, n}, n3l, &c.

n sin 6, cos s i n e 2 c o s ^1 L L= L + + . + & C . ,4 26,-fsine, 2e2 + sine2 2e3 + sme3

an equation which holds for all values of y included between0 and I, and consequently for all those which are included between0 and — I, when x = 0.

Substituting the known values of a19 bl9 a2, b2, a3, b3, &c. inthe general value of v, we have the following equation, whichcontains the solution of the proposed problem,

v s i nn jcosy /sinnjcosn,y y^r^i &

4.4 2 V + sin 2nJ \2nxl + sin 2w^

/sinnjcosnj ^ y g ^ i & \\ 2^2 + sin 2w^ 7

sinVcos^ /s in^cos ¥ ^ ^ t \2 3Z + sin 2n3l \2nJ, + sin 2nxl ' 7

+ &c , (E).

The quantities denoted by nl9 n29 n3, &c. are infinite in

number, and respectively equal to the quantities j , ~, * &c.;e t c

the arcs, e1? e2, e3, &c, are the roots of the definite equationhi

e tan e = -j-.

326. The solution expressed by the foregoing equation E isthe only solution which belongs to the problem; it represents the

i • i i D xi x- $*v d2v d2v ~, . , . , .

general integral of the equation -pl + -—2 + - -2 = 0, in which thearbitrary functions have been determined from the given condi-tions. It is easy to see that there can be no different solution.In fact, let us denote by ty(x9 y, z) the value of v derived from theequation (JE), it is evident that if we gave to the solid initial tem-peratures expressed by yfr(x, y, z)9 no change could happen in thesystem of temperatures, provided that the section at the originwere retained at the constant temperature 1: for the equationd2v d2v dzv n . . ,. „ 1 ±1 . ,3~a + ^~2 + j~a = 0 being satisfied, the instantaneous variation of


the temperature is necessarily nothing. The same would not bethe case, if after having given to each point within the solid whoseco-ordinates are x, y, z the initial temperature y}r(x, y, z\ we gaveto all points of the section at the origin the temperature 0. Wesee clearly, and without calculation, that in the latter case thestate of the solid would change continually, and that the originalheat which it contains would be dissipated little by little into theair, and into the cold mass which maintains the end at the tem-perature 0. This result depends on the form of the functionyfr(x, y, z), which becomes nothing when x has an infinite value asthe problem supposes.

A similar effect would exist if the initial temperatures insteadof being + yjr (x, y, z) were — ty (#, y, z) at all the internal pointsof the prism; provided the section at the origin be maintainedalways at the temperature 0. In each case, the initial tempera-tures would continually approach the constant temperature of themedium, which is 0; and the final temperatures would all be nul.

327. These preliminaries arranged, consider the movement ofheat in two prisms exactly equal to that which was the subject ofthe problem. For the first solid suppose the initial temperaturesto be + ^(x, y, z), and that the section at origin A is maintainedat the fixed temperature 1. For the second solid suppose theinitial temperatures to be — ^(xy y, z), and that at the origin Aall points of the section are maintained at the temperature 0. Itis evident that in the first prism the system of temperatures can-not change, and that in the second this system varies continuallyup to that at which all the temperatures become nul.

If now we make the two different states coincide in the samesolid, the movement of heat is effected freely, as if each systemalone existed. In the initial state formed of the two unitedsystems, each point of the solid has zero temperature, except thepoints of the section A} in accordance with the hypothesis. Nowthe temperatures of the second system change more and more,and vanish entirely, whilst those of the first remain unchanged.Hence after an infinite time, the permanent system of tempera-tures becomes that represented by equation Ey or v = ty{x, y, z).It must be remarked that this result depends on the conditionrelative to the initial state; it occurs whenever the initial heat

CHAP. VII.] GEOMETRICAL CONSTRUCTION. 317

contained in the prism is so distributed, that it would vanishentirely, if the end A were maintained at the temperature 0.

328. We may add several remarks to the preceding solution.hi

1st, it is easy to see the nature of the equation e tan e = -r; we

need only suppose (see fig. 15) that we have constructed the curveu = e tan e, the arc e being taken for abscissa, and u for ordinate.The curve consists of asymptotic branches.

Fig. 15.

\Z2

The abscissae which correspond to the asymptotes are ^TT,

3 5 7^7r, ^TT, »7T, &c.; those which correspond to points of mtersec-22 Z A

tion are 7r, 27r, 37r, &C. If now we raise at the origin an ordinate

equal to the known quantity -j-, and through its extremity draw

a parallel to the axis of abscissae, the points of intersection will

give the roots of the proposed equation e tan e = j - . The con-

struction indicates the limits between which each root lies. We

shall not stop to indicate the process of calculation which must be

employed to determine the values of the roots. Eesearches of

this kind present no difficulty.

329. 2nd. We easily conclude from the general equation (E)that the greater the value of x becomes, the greater that term of

the value of v becomes, in which we find the fraction e~Wwi2+*12,with respect to each of the following terms. In fact, nl9 n2, w3,

&c. being increasing positive quantities, the fraction e~Xx/2n{Z is

318 THEOKY OF HEAT. [CHAP. VII.

greater than any of the analogous fractions which enter into thesubsequent terms.

Suppose now that we can observe the temperature of a pointon the axis of the prism situated at a very great distance x, andthe temperature of a point on this axis situated at the distancex + 1 , 1 being the unit of measure; we have then y = 0, z = 0,and the ratio of the second temperature to the first is sensiblyequal to the fraction e~ v^2%2. This value of the ratio of the tem-peratures at the two points on the axis becomes more exact as thedistance x increases.

It follows from this that if we mark on the axis points each ofwhich is at a distance equal to the unit of measure from the pre-ceding, the ratio of the temperature of a point to that of the point

which precedes it, converges continually to the fraction e~^2ni'2;thus the temperatures of points situated at equal distances endby decreasing in geometrical progression. This law always holds,whatever be the thickness of the bar, provided we consider pointssituated at a great distance from the source of heat.

It is easy to see, by means of the construction, that if thequantity called I, which is half the thickness of the prism, is verysmall, n1 has a value very much smaller than n2> or ns, &c.; it

follows from this that the first fraction e~x^2ni2 is very muchgreater than any of the analogous fractions. Thus, in the case inwhich the thickness of the bar is very small, it is unnecessary tobe very far distant from the source of heat, in order that the tem-peratures of points equally distant may decrease in geometricalprogression. The law holds through the whole extent of the bar.

330. If the half thickness I is a very small quantity, thegeneral value of v is reduced to the first term which contains

e~ 2%2. Thus the function v which expresses the temperature ofa point whose co-ordinates are x, y, and s, is given in this case bythe equation

/ 4sinrcZ \2 -X\Twv = in-,—r~in c o s ny c o s nz e XSim\2nl + sin 2nlJ J '

the arc e or nl becomes very small, as we see by the construction.

The equation e tan e = T reduces then to e2 = T ; the first value of

CHAP. VII.] CASE OF A THIN BAR. 319

e, or ev is A / T T ; by inspection of the figure we know the values ofV AC

the other roots, so that the quantities e e2, e8, e4, e5, &c. are the

following /v/-r-, w, 27r, 37r, 47r,&c. The values of nv w2, w3, n4, w5, &c.are, therefore,

1 /h 7T 2TT 3TT ,

VZV &' 7' T ' T ' ;

whence we conclude, as was said above, that if I is a very smallquantity, the first value n is incomparably greater than all theothers, and that we must omit from the general value of v all theterms which follow the first. If now we substitute in the firstterm the value found for n, remarking that the arcs nl and 2nl areequal to their sines, we have

(y lhl\ (z lhl\ _x /§

the factor A / -r- which enters under the symbol cosine being very

small, it follows that the temperature varies very little, fordifferent points of the same section, when the half thickness I isvery small. This result is so to speak self-evident, but it is usefulto remark how it is explained by analysis. The general solutionreduces in fact to a single term, by reason of the thinness of thebar, and we have on replacing by unity the cosines of very small

arcs v = e *Vw , an equation which expresses the stationary tempe-ratures in the case in question.

We found the same equation formerly in Article 76; it isobtained here by an entirely different analysis.

331. The foregoing solution indicates the character of themovement of heat in the interior of the solid. It is easy to seethat when the prism has acquired at all its points the stationarytemperatures which we are considering, a constant flow of heatpasses through each section perpendicular to the axis towards theend which was not heated. To determine the quantity of flowwhich corresponds to an abscissa x, we must consider that thequantity which flows during unit of time, across one element of


the section, is equal to the product of the coefficient k, of the areaft 1}

dydz, of the element dt, and of the ratio -=- taken with the nega-dx

/* /» 7

tive sign. We must therefore take the integral -kldyldzj-,

from z = 0 to z — I, the half thickness of the bar, and then fromy = 0 to y = I. We thus have the fourth part of the whole flow.

The result of this calculation discloses the law according towhich the quantity of heat which crosses a section of the bardecreases; and we see that the distant parts receive very littleheat from the source, since that which emanates directly from itis directed partly towards the surface to be dissipated into the air.That which crosses any section whatever of the prism forms, if wemay so say, a sheet of heat whose density varies from one pointof the section to another. It is continually employed to replacethe heat which escapes at the surface, through the whole end ofthe prism situated to the right of the section: it follows thereforethat the whole heat which escapes during a certain time from thispart of the prism is exactly compensated by that which penetratesit by virtue of the interior conducibility of the solid.

To verify this result, we must calculate the produce of the flowestablished at the surface. The element of surface is dxdy> and vbeing its temperature, hvdxdy is the quantity of heat whichescapes from this element during the unit of time. Hence the

integral h jdxldyv expresses the whole heat which has escaped

from a finite portion of the surface. We must now employ theknown value of v in y} supposing z = I, then integrate once fromy = 0 to y = ly and a second time from x = x up to x = oo . Wethus find half the heat which escapes from the upper surface ofthe prism; and taking four times the result, we have the heat lostthrough the upper and lower surfaces.

If we now make use of the expression h \dx\dzv, and give to

y in v its value I, and integrate once from z = 0 to z = I, and asecond time from x = 0 to x = oo ; we have one quarter of the heatwhich escapes at the lateral surfaces.

The integral h I dx jdy v} taken betweenthelimits indicated gives

CHAP. VII.] HEAT LOST AND TRANSMITTED. 321

ha

— sin ml cos nl e~x*m*+n\

and the integral h \dx\dzv gives

ha: cos ml Q'mnle'x>s/m2'jrn\

Hence the quantity of heat which the prism loses at its surface,throughout the part situated to the right of the section whoseabscissa is x, is composed of terms all analogous to

I g-Wm*+»» I __ g i n mlcog nl _|_ _ c o g ml g i n nA t

Jm' + n2 [m n )On the other hand the quantity of heat which during the sametime penetrates the section whose abscissa is x is composed ofterms analogous to

mn

the following equation must therefore necessarily hold

. 7 . 7 h 1 7

sin mi sin nl = . sin ml cos nlm Jm2 + n2

j c o s ml sin nl,j T =n Jm + n2

or h (m2 + n2) sin ml sin nl = hm cos ml sin nl + hn sin ml cos n£:

now we have separately,

km2 sin ml cos w£ = Am cos ml sin ??£,

m sin ml hcos ra£ A;

we have also

lew? sin ?zi sin mZ = hn cos n? sin ml,.

n sin nZ AQY —- = —

k 'Hence the equation is satisfied. This compensation which is in-cessantly established between the heat dissipated and the heattransmitted, is a manifest consequence of the hypothesis; andanalysis reproduces here the condition which has already been ex-

F. H. 21

322 THEORY OF HEAT. [CHAP. VII,

pressed; but it was useful to notice this conformity in a newproblem, which had not yet been submitted to analysis.

332. Suppose the half side I of the square which serves as thebase of the prism to be very long, and that we wish to ascertain thelaw according to which the temperatures at the different points ofthe axis decrease; we must give to y and z nul values in thegeneral equation, and to I a very great value. Now the construe-

tion shews in this case that the first value of € is TT , the secondz

-=-, the third —, &c. Let us make these substitutions in the general

HT O77"

equation, and replace nxl, n2l, n3l, n4l, &c. by their values ^ , — ,

-^,-£, and also substitute the fraction a for e'1^; we then find

v g ) = 1 (a^2 - | a V I ^ + \ aVlW2 - &c.)

-&c .

We see by this result that the temperature at different pointsof the axis decreases rapidly according as their distance from theorigin increases. If then we placed on a support heated andmaintained at a permanent temperature, a prism of infinite height,having as base a square whose half side I is very great; heat wouldbe propagated through the interior of the prism, and would be dis-sipated at the surface into the surrounding air which is supposedto be at temperature 0. When the solid had arrived at a fixedstate, the points of the axis would have very unequal tempera-tures, and at a height equal to half the side of the base thetemperature of the hottest point would be less than one fifth partof the temperature of the base.

CHAPTEE VIII.

OF THE MOVEMENT OF HEAT IN A SOLID CUBE.

333. IT still remains for us to make use of the equation

dv _dt

which represents the movement of heat in a solid cube exposedto the action of the air (Chapter II., Section V.). Assuming, inthe first place, for v the very simple value e~mtcosnx cospy cos qz,if we substitute it in the proposed equation, we have the equa-tion of condition m = k (n2 +p2 + q2), the letter k denoting the

rr

coefficient - F ^ . It follows from this that if we substitute for

n, p, q any quantities whatever, and take for m the quantityJc(n2 + p2 + q2), the preceding value of v will always satisfy thepartial differential equation. We have therefore the equationv = e^b^+v2*^*cosnxcospy cos qz. The nature of the problemrequires also that if x changes sign, and if y and z remain thesame, the function should not change; and that this should alsohold with respect to y or z: now the value of v evidently satisfiesthese conditions.

334. To express the state of the surface, we must employ thefollowing equations:

dy^"-""" ' ^ -

21—2

324 THEORY OF HEAT. [CHAP. VIII.

These ought to be satisfied when x = ± a, or y = ± a, or z = ± a.The centre of the cube is taken to be the origin of co-ordinates :and the side is denoted by a.

The first of the equations (b) gives

+ e~mt n sin nx cos py cos qz + jr cos nx cos py cos qz = 0,

or + ntann# + ^ = 0 ,A

an equation which must hold when x = ± a.

It follows from this that we cannot take any value what-ever for n, but that this quantity must satisfy the condition

nata,nna = a. We must therefore solve the definite equationK

€ tan e = -^a, which gives the value of e, and take n = - . Now the

equation in e has an infinity of real roots; hence we can find forn an infinity of different values. We can ascertain in the samemanner the values which may be given to p and to q; they areall represented by the construction which was employed in thepreceding problem (Art. 321). Denoting these roots by niyn2,n^ &c.;we can then give to v the particular value expressed by theequation

provided we substitute for n one of the roots nlt n2, nz> &c, andselect p and q in the same manner.

335. We can thus form an infinity of particular values of v,and it evident that the sum of several of these values will alsosatisfy the differential equation (a), and the definite equations (6).In order to give to v the general form which the problem requires,we may unite an indefinite number of terms similar to the term

c o s nx C0Spy cos qz.

The value of v may be expressed by the following equation:

v = (ax cos nxoo e-]cn& + a2 cos n2x e~kn*H + a3 cos nzxe~kn*H + &c),

(bi cos nxy e~kniH + b2 cos n2y e~kn^ + bB cos n3y e~kn*H + &c),

(ct cos nxze-kn^H + c 2 c o s n 2 z e r ^ ^ + cQcosnzy #-*»»** + &c.).

CHAP. VIII.] GENERAL VALUE OF V. 325

The second member is formed of the product of the threefactors written in the three horizontal lines, and the quantitiesa19 a2, a3, &c. are unknown coefficients. Now, according to thehypothesis, if t be made = 0, the temperature must be the same atall points of the cube. We must therefore determine at, a2> a3> &c,so that the value of v may be constant, whatever be the values ofxy y, and s, provided that each of these values is included betweena and — a. Denoting by 1 the initial temperature at all points ofthe solid, we shall write down the equations (Art. 323)

1 = ax cos nxx + a2 cos n2x + a3 cos n3x + &c,

1 = bx cos nxy + 6a cos n2y + b3 cos n3y + &c,

1 = cx cos nxz + c2 cos n2z + c8 cos n3z + &c,

in which it is required to determine av a2, a3, &c. After multi-plying each member of the first equation by cosnx, integratefrom x = 0 to x = a: it follows then from the analysis formerlyemployed (Art. 324) that we have the equation

sin nn cos n,x sin noa cos nox sin noa cos nox""1 / - s in2^a\ 1 / , sin 2n2a\ 1 /- sin 2nji\1 / ,

2 f \"2 x V 2^(1 / 2 f \ " 2?i2a / 2 3 V 2n3a

+ &c.

~ ,. , xl ... 1 / - , sin2w/A ,Denoting by ^ the quantity ^ m — ^ — — I , we have

sinw-a sinwoa sin njz 0

1 = i- cos w. H - cos 7i2 + ^-cos njx + &c.

This equation holds always when we give to a? a value includedbetween a and — a.

From it we conclude the general value of v, which is given bythe following equation

cos ^ tf-*<«+&c.)

c o s nje-**" + S ^ - cos nj, e'^ SlU nafi **

336. The expression for v is therefore formed of three similarfunctions, one of x, the other of y, and the third of z, which iseasily verified directly.

In fact, if in the equation

dh c

we suppose v = XYZ; denoting by X a function of x and t,by Y a function of y and t, and by Z a function of z and t, we have

dz )

which implies the three separate equations

_ CPY dZ_~ df' dt~h

dt~dx*' dt~ df' dtWe must also have as conditions relative to the surface,

^ + i 7 = 0 ^T+AF-O ^ + F - Odx K ay K dz K '

whence we deducedX+hY-0 dY+hY-fi dZ+h7 A

It follows from this, that, to solve the problem completely, it is

enough to take the equation ^ - = Ar — , and to add to it the

equation of condition -j~ -f ^u = 0, which must hold when x = a.

We must then put in the place of xy either y or z, and we shallhave the three functions X, Yy Z> whose product is the generalvalue of v.

Thus the problem proposed is solved as follows:

v = </> (x, t)4> (y, t) <j> 0 , t);

, . N sin nxi hr) 2/ sin n2a . g

d> (x. t) = cos n.x e kUlt -\ cos nja e lm*H

sin na Jrn 2. 04 cos njc e-Jcn*H + &c.;

CHAP. VIII.] ONE SOLUTION ONLY. 327

nv n2, ns) &c. being given by the following equation

ha€ tan e = -jr ,

Jti.

in which e represents na and the value of /j,t is

1 /2 V )2 V 2nfl ) '

In the same manner the functions <p (y, t), <fr (z, t) are found.

337. We may be assured that this value of v solves the pro-blem in all its extent, and that the complete integral of the partialdifferential equation (a) must necessarily take this form in orderto express the variable temperatures of the solid.

In fact, the expression for v satisfies the equation (a) and theconditions relative to the surface. Hence the variations of tempe-rature which result in one instant from the action of the moleculesand from the action of the air on the surface, are those which weshould find by differentiating the value of v with respect to thetime t It follows that if, at the beginning of any instant, thefunction v represents the system of temperatures, it will stillrepresent those which hold at the commencement of the followinginstant, and it may be proved in the same manner that the vari-able state of the solid is always expressed by the function v, inwhich the value of t continually increases. Now this functionagrees with the initial state: hence it represents all the laterstates of the solid. Thus it is certain that any solution whichgives for v a function different from the preceding must be wrong.

338. If we suppose the time t, which has elapsed, to havebecome very great, we no longer have to consider any but thefirst term of the expression for v; for the values nv n2, n3, &c. arearranged in order beginning with the least. This term is givenby the equation

/sinn.av = I L- cos n.x cos n,y cos n,z

this then is the principal state towards which the system of tem-peratures continually tends, and with which it coincides withoutsensible error after a certain value of t In this state the tempe-


rature at every point decreases proportionally to the powers ofthe fraction e~zh7ll*\ the successive states are then all similar, orrather they differ only in the magnitudes of the temperatureswhich all diminish as the terms of a geometrical progression, pre-serving their ratios. We may easily find, by means of the pre-ceding equation, the law by which the temperatures decrease fromone point to another in direction of the diagonals or the edges ofthe cube, or lastly of a line given in position. We might ascer-tain also what is the nature of the surfaces which determine thelayers of the same temperature. We see that in the final andregular state which we are here considering, points of the samelayer preserve always equal temperatures, which would not holdin the initial state and in those which immediately follow it.During the infinite continuance of the ultimate state the mass isdivided into an infinity of layers all of whose points have a com-mon temperature.

339. It is easy to determine for a given instant the meantemperature of the mass, that is to say, that which is obtained bytaking the sum of the products of the volume of each moleculeby its temperature, and dividing this sum by the whole volume.

We thus form the expression \\\—-^~z—, which is that of the

mean temperature V. The integral must be taken successivelywith respect to x, y, and zy between the limits a and — a: v beingequal to the product X YZy we have

thus the mean temperature is f I——j , since the three complete

integrals have a common value, hence

3fT? / s i n n , a\21 , 2, / s i n no a\21 . . . o

The quantity na is equal to e? a root of the equation e tan € = -^ ,

and /t is equal to x (1 H ^— j . We have then, denoting the

different roots of this equation by ev e2, e8, &c,

CHAP. VIII.] CUBE AND SPHEEE COMPARED. 329

siu 2et V e2 / sin

T +

6j is between 0 and - 7r, e2 is between ir and — , e3 between 2ir and

- IT, the roots e2, e3, e4, &c. approach more and more nearly to the

inferior limits 7r, 2-7T, Sir, &c, and end by coinciding with themwhen the index i is very great. The double arcs 2ev 2e2, 2e3, &c,are included between 0 and 7r, between 2TT and 3TT, between 47r

.and 57r; for which reason the sines of these arcs are all positive :, ,.,. i sin2e1 - sin269 0 . . . , . , , 1

the quantities 1 H—^—l, 1 H—^—z, &c, are positive and included1 2

between 1 and 2. I t follows from this that all the terms whichenter into the value of lj V are positive.

340. We propose now to compare the velocity of cooling inthe cube, with that which we have found for a spherical mass.We have seen that for either of these bodies, the system of tem-peratures converges to a permanent state which is sensibly attainedafter a certain time; the temperatures at the different points ofthe cube then diminish all together preserving the same ratios,and the temperatures of one of these points decrease as the termsof a geometric progression whose ratio is not the same in the twobodies. It follows from the two solutions that the ratio for the

e2

sphere is e~kn% and for the cube e~*&k. The quantity n is given bythe equation

cos na - h1na = 1 —^ay

sin na Ka being the semi-diameter of the sphere, and the quantity e is given

by the equation e tan e = -j^a, a being the half side of the cube.

This arranged, let us consider two different cases; that inwhich the radius of the sphere and the half side of the cube areeach equal to a, a very small quantity; and that in which thevalue of a is very great. Suppose then that the two bodies are of

^small dimensions; -^having a very small value, the same is the

-^case with e, we have therefore-^ = e2, hence the fraction

e a2 is equal to e cm.

Thus the ultimate temperatures which we observe are expressed in, _J*M T/1 , ,. wacosTia ^ &

the form J_e £#». If now m the equation —: = 1 — -^a, we1 sm na K

suppose the second member to differ very little from unity, we find

•«.= -^-, hence the fraction e is e C!#*.

We conclude from this that if the radius of the sphere is very-small, the final velocities of cooling are the same in that solid andin the circumscribed cube, and that each is in inverse ratio of theradius ; that is to say, if the temperature of a cube whose half sideis a passes from the value A to the value B in the time t> a spherewhose semi-diameter is a will also pass from the temperature Ato the temperature B in the same time. If the quantity a werechanged for each body so as to become a\ the time required forthe passage from A to B would have another value t\ and theratio of the times t and t' would be that of the half sides a and a.The same would not be the case when the radius a is very great:for e is then equal to ^7r, and the values of na are the quantities7T, 27T, 37T, 47T, &C.

We may then easily find, in this case, the values of the frac-~Ze-k _7 . 2 __^!L2 _ ^ 2

tions e <*>* , e ; they are e ^ and e a*.

From this we may derive two remarkable consequences: 1st, whentwo cubes are of great dimensions, and a and a are their half-sides ; if the first occupies a time t in passing from the temperatureA to the temperature By and the second the time t' for the sameinterval; the times t and t' will be proportional to the squares a2

and a2 of the half-sides. We found a similar result for spheres ofgreat dimensions. 2nd, If the length a of the half-side of a cubeis considerable, and a sphere has the same magnitude a for radius,and during the time t the temperature of the cube falls from A toBj a different time tr will elapse whilst the temperature of the

CHAP. VIII.] REMARKS. 331

sphere is falling from A to j?, and the times t and tf are in theratio of 4 to 3.

Thus the cube and the inscribed sphere cool equally quicklywhen their dimension is small; and in this case the duration ofthe cooling is for each body proportional to its thickness. If thedimension of the cube and the inscribed sphere is great, the finalduration of the cooling is not the same for the two solids. Thisduration is greater for the cube than for the sphere, in the ratio of4 to 3, and for each of the two bodies severally the duration of thecooling increases as the square of the diameter.

341. We have supposed the body to be cooling slowly in at-mospheric air whose temperature is constant. We might submitthe surface to any other condition, and imagine, for example, thatall its points preserve, by virtue of some external cause, the fixedtemperature 0. The quantities n, %), q, which enter into the valueof v under the symbol cosine, must in this case be such that cos nxbecomes nothing when x has its complete value a, and that thesame is the case with cos py and cos qz. If 2a the side of thecube is represented by iry 2TT being the length of the circumferencewhose radius is 1; we can express a particular value of v by thefollowing equation, which satisfies at the same time the generalequation of movement of heat, and the state of the surface,

v = e CD cos x. cos y. cos z.

This function is nothing, whatever be the time t, when x or y or zTT TT

receive their extreme values 4- •= or — ^ : but the expression for the

temperature cannot have this simple form until after a consider-able time has elapsed, unless the given initial state is itselfrepresented by cos x cos y cos z. This is what we have supposedin Art. 100, Sect. Vlli. Chap. I. The foregoing analysis proves thetruth of the equation employed in the Article we have j ust cited.

Up to this point we have discussed the fundamental problemsin the theory of heat, and have considered the action of thatelement in the principal bodies. Problems of such kind and orderhave been chosen, that each presents a new difficulty of a higherdecree. We have designedly omitted a numerous variety of


intermediate problems, such as the problem of the linear movementof heat in a prism whose ends are maintained at fixed temperatures,or exposed to the atmospheric air. The expression for the variedmovement of heat in a cube or rectangular prism which is coolingin an aeriform medium might be generalised, and any initialstate whatever supposed. These investigations require no otherprinciples than those which have been explained in this work.

A memoir was published by M. Fourier in the Memoires de VAcademie desSciences, Tome vn. Paris, 1827, pp. 605—624, entitled, Memoire sur la distinction desracines imaginaires, et sur V application des theorhnes ^analyse algebrique auxequations transcendantes qui dependent de la theorie de la chaleur. It contains aproof of two propositions in the theory of heat. If there be two solid bodies ofsimilar convex forms, such that corresponding elements have the same density,specific capacity for heat, and conductivity, and the same initial distribution oftemperature, the condition of the two bodies will always be the same after timeswhich are as the squares of the dimensions, when, 1st, corresponding elementsof the surfaces are maintained at constant temperatures, or 2nd, when the tem-peratures of the exterior medium at corresponding points of the surface remainconstant.

For the velocities of flow along lines of flow across the terminal areas s, sf ofcorresponding prismatic elements are as u - v: ur - v't where (u, v), (u\ v') are tem-peratures at pairs of points at the same distance -J A on opposite sides of s and s';and if n : nr is the ratio of the dimensions, u-v : u'-v'=n':n. If then, dt, dt'becorresponding times, 'the quantities of heat received by the prismatic elements areas sk (u -v)dt : s'h (u' - v') dt\ or as n2n'dt : n2'ndt\ But the volumes being asn?1 n'*, if the corresponding changes of temperature are always equal we must have

nWdt _ ri2ndtf dt _ w2

riA = nri '01Cdi' = rif*'

In the second case we must suppose E: H'=n': n. [A. F.]

CHAPTER IX.

OF THE DIFFUSION OF HEAT.

FIRST SECTION.

Of the free movement of heat in an infinite line.

342. HERE we consider the movement of heat in a solidhomogeneous mass, all of whose dimensions are infinite. Thesolid is divided by planes infinitely near and perpendicular to acommon axis; and it is first supposed that one part only of thesolid has been heated, that, namely, which is enclosed betweentwo parallel planes A and By whose distance is g; all other partshave the initial temperature 0; but any plane included betweenA and B has a given initial temperature, regarded as arbitrary,and common to every point of the plane; the temperature is dif-ferent for different planes. The initial state of the mass beingthus defined, it is required to determine by analysis all the suc-ceeding states. The movement in question is simply linear, andin direction of the axis of the plane ; for it is evident that therecan be no transfer of heat in any plane perpendicular to the axis,since the initial temperature at every point in the plane is thesame.

Instead of the infinite solid we may suppose a prism of verysmall thickness, whose lateral surface is wholly impenetrable toheat. The movement is then considered only in the infinite linewhich is the common axis of all the sectional planes of the prism.

The problem is more general, when we attribute temperaturesentirely arbitrary to all points of the part of the solid which has

334 THEORY OF HEAT. [CHAP. IX.

been heated, all other points of the solid having the initial tem-perature 0. The laws of the distribution of heat in an infinitesolid mass ought to have a simple and remarkable character;since the movement is not disturbed by the obstacle of surfaces,or by the action of a medium.

343. The position of each point being referred to three rect-angular axes, on which we measure the co-ordinates x, y, z, thetemperature sought is a function of the variables x, y, z, and ofthe time t. This function v or <f>(x,y9 z, t) satisfies the generalequation

dv __ K (d2v d2v d2v^^m)\d2+df + d?

Further, it must necessarily represent the initial state which isarbitrary; thus, denoting by F{x, y, z) the given value of thetemperature at any point, taken when the time is nothing, that isto say, at the moment when the diffusion begins, we must have

Hence we must find a function v of the four variables x, y, z, t,which satisfies the differential equation (a) and the definite equa-tion (b).

In the problems which we previously discussed, the integral issubject to a third condition which depends on the state of thesurface: for which reason the analysis is more complex, and thesolution requires the employment of exponential terms. Theform of the integral is very much more simple, when it need onlysatisfy the initial state; and it would be easy to determine atonce the movement of heat in three dimensions. But in order toexplain this part of the theory, and to ascertain according to whatlaw the diffusion is effected, it is preferable to consider first thelinear movement, resolving it into the two following problems : weshall see in the sequel how they are applied to the case of threedimensions.

344. First problem: a part ah of an infinite line is raised atall points to the temperature 1; the other points of the line are atthe actual temperature 0; it is assumed that the heat cannot bedispersed into the surrounding medium; we have to determine

SECT. I.] TWO PROBLEMS. 335

what is the state of the line after a given time. This problemmay be made more general, by supposing, 1st, that the initialtemperatures of the points included between a and b are unequaland represented by the ordinates of any line whatever, which weshall regard first as composed of two symmetrical parts (see fig. 16);

Fig. 16.

2nd, that part of the heat is dispersed through the surface of thesolid, which is a prism of very small thickness, and of infinitelength.

The second problem consists in determining the successivestates of a prismatic bar, infinite in length, one extremity ofwhich is submitted to a constant temperature. The solution ofthese two problems depends on the integration of the equation

dv K d2v HLdt CD dx2 CDSV>

(Article 105), which expresses the linear movement of heat, v isthe temperature which the point at distance x from the originmust have after the lapse of the time t; K, H, G, D, L, S, denotethe internal and surface conducibilities, the specific capacity forheat, the density, the contour of the perpendicular section, andthe area of this section.

345. Consider in the first instance the case in which heat ispropagated freely in an infinite line, one part of which ab hasreceived any initial temperatures; all other points having theinitial temperature 0. If at each point of the bar we raise theordinate of a plane curve so as to represent the • actual tempera-ture at that point, we see that after a certain value of the time t,the state of the solid is expressed by the form of the curve.Denote by v = F(x) the equation which corresponds to the giveninitial state, and first, for the sake of making the investigation

more simple, suppose the initial form of the curve to be composedof two symmetrical parts, so that we have the condition

F(x)=F(-x).

J,et CD- /c} Qpg-K,

in the equation -7- = k -j—2 — hv, make v = erht u, and we have

du _ , d2u~dt ~ dz*'

Assume a particular value of u, namely, a cos qx e~kqH; a and qbeing arbitrary constants. Let qv q2) q3, &c. be a series of anyvalues whatever, and al5 <z2, a3, &c. a series of correspondingvalues of the coefficient Q, we have

u = ax cos (qjc) e~kqi2t + a2 cos (q2cs) e-kq*H + aQ cos (qsx) e-kq^ + &c.

Suppose first that the values qv q2, q3, &c. increase by infinitelysmall degrees, as the abscissae q of a certain curve; so that theybecome equal to dq, 2dqf 3dq, &c.; dq being the constant differen-tial of the abscissa; next that the values av a2> a3, &c. are pro-portional to the ordinates Q of the same curve, and that theybecome equal to Qxdq, Q2dq, Q3dq, &c, Q being a certain functionof q. It follows from this that the value of u may be expressedthus:

u = Idq Q cos qx e~kqH,

Q is an arbitrary function f(q), and the integral may be takenfrom 2 = 0 to £=00. The difficulty is reduced to determiningsuitably the function Q.

346. To determine Q, we must suppose t = 0 in the expressionfor u, and equate u to F (x). We have therefore the equation ofcondition

F(x) = jdqQcos qx.

If we substituted for Q any function of q, and conducted theintegration from q = 0 to q = 00, we should find a function of x:it is required to solve the inverse problem, that is to say, toascertain what function of q, after being substituted for Q, givesas the result the function Fix), a remarkable problem whosesolution demands attentive examination.

SECT. I.J AN INVERSE PROBLEM. 337

Developing the sign of thQ integral, we write as follows, theequation from which the value of Q must be derived:

F(x) =• dq Q1 cos qxx + dqQ2 cos q2x + dqQ3 cos qzx + &c.

In order to make all the terms of the second member dis-appear, except one, multiply each side by dxcosrx, and thenintegrate with respect to x from x = 0 to x = rnr, where n is aninfinite number, and r represents a magnitude equal to any oneof qlf q2, qzy &c, or which is the same thing dq, 2dq, 3dq, &c. Letqi be any value whatever of the variable q, and qj another value,namely, that which we have taken for r; we shall have r =jdq,and q = idq. Consider then the infinite number n to express howmany times unit of length contains the element dq, so that we

have n = -j- . Proceeding to the integration we find that the

value of the integral jdx cosqx cos rx is nothing, whenever r and

q have different magnitudes; but its value is - niry when q = r.

This follows from the fact that integration eliminates from thesecond member all the terms, except one ; namely, that whichcontains qj or r. The function which affects the same termis Qj, we have therefore

/ •dx F (x) cos qx — dq Q3 ^ nir,

'2

and substituting for ndq its value 1, we have

— —— = / CLX JO \X) LOr> GOU.£ J

We find then, in general, ^ ^ = I dx F (x) cos qx. Thus, to2 Jo

determine the function Q which satisfies the proposed condition,we must multiply the given function F(x) by cfocos^, and in-

2tegrate from x nothing to x infinite, multiplying the result by - ;

that is to say, from the equation F(x) = jdqf{q) cos qx, we deduce

f (q) =- dxF(x) cos q.r, the function F (r) representing the

F ii. 22

initial temperatures of an infinite prism, of which an intermediatepart only is heated. Substituting the value off(q) in the expres-sion for F (x), we obtain the general equation

-F(x)=\ dqcosqxj dxF(x)cosqx (e).2i Jo Jo

347. If we substitute in the expression for v the value whichwe have found for the function Q, we have the following integral,which contains the complete solution of the proposed problem,

= e-u I dqcosqxe~kqH jdxF(x) cos qx.

The integral, with respect to x, being taken from x nothingto x infinite, the result is a function of q; and taking then theintegral with respect to q from q = 0 to q = oo, we obtain for v afunction of x and t, which represents the successive states of thesolid. Since the integration with respect to x makes this variabledisappear, it may be replaced in the expression of v by any varia-ble a, the integral being taken between the same limits, namelyfrom a = 0 to a = oo . We have then

— = e~u I dq cos qx e~kqH \ dx F(a) cos q-x>£ Jo Jo

/.CO /»OO

' _ e-u I fa F(a) dq prk<iH cos qx cos qx,* Jo Jo

17V

or

The integration with respect to q will give a function of x9

t and a, and taking the integral with respect to a we find a func-tion of x and t only. In the last equation it would be easy toeffect the integration with respect to q, and thus the expressionof v would be changed. We can in general give different formsto the integral of the equation

dr . d2v 7

fit dx2

they all represent the same function of x and t

348. Suppose in the first place that all the initial tempera-tures of points included between a and b, from x = — 1, to x = 1,have the common value 1, and that the temperatures of all the

SECT. I.] FUNCTIONS EXPRESSED BY INTEGRALS. 339

other points are nothing, the function F(x) will be given by thiscondition. It will then be necessary to integrate, with respect tox} from x = 0 to x = 1, for the rest of the integral is nothingaccording to the hypothesis. We shall thus find

Q = - ^ and ^ - 6"« C^L e-W cos qx sin q.IT q 2 Jo q i *

The second member may easily be converted into a convergentseries, as will be seen - presently; it represents exactly the stateof the solid at a given instant, and if we make in it t = 0, it ex-presses the initial state.

Thus the function — I — sin q cos qx is equivalent to unity, if

we give to x any value included between — 1 and 1: but thisfunction is nothing if to x any other value be given not includedbetween — 1 and 1. We see by this that discontinuous functionsalso may be expressed by definite integrals.

349. In order to give a second application of the precedingformula, let us suppose the bar to have been heated at one of itspoints by the constant action of the same source of heat, andthat it has arrived at its permanent state which is known to berepresented by a logarithmic curve.

It is required to ascertain according to what law the diffusionof heat is effected after the source of heat is withdrawn. Denoting

o

by F (x) the initial value of the temperature, we shall have

F(x)=Ae KS; A is the initial temperature of the pointmost heated. To simplify the investigation let us make -4 = 1,

and -TF- 7 =1 . We have then F(x)=e~x, whence we deduceKb

Q = / dwe~x cosqx, and taking the integral from x nothing to x

infinite, Q=^ 2* Thus the value of v in x and t is given by.

the following equation:

rdq^osqx

22—2

340 THEORY. OF HEAT. [CHAP. JX.

3oO. If we make t = 0, we have — = / ^ — — \ - \ which cor-2 Jo J- + 5

responds to the initial state. Hence the expression - I - y - r * -

is equal to e-x. It must be remarked that the function F(x),which represents the initial state, does not change its value accord-ing to hypothesis when x becomes negative. The heat communi-cated by the source before the initial state was formed, ispropagated equally to the right and the left of the point 0, whichdirectly receives it: it follows that the line whose equation is

y=—\ -i {- is composed of two symmetrical branches which

are formed by repeating to right and left of the axis of y the partof the logarithmic curve which is on the right of the axis of y, andwhose equation is y = e~x. We see here a second example of adiscontinuous function expressed by a definite integral. This

function - I ~f %- is equivalent to e~x when x is positive, butir J o 1 H- q

it is ex when x is negative1.

351. The problem of the propagation of heat in an infinitebar, one end of which is subject to a constant temperature, isreducible, as we shall see presently, to that of the diffusion of heatin an infinite line; but it must be supposed that the initial heat,instead of affecting equally the two contiguous halves of the solid,is distributed in it in contrary manner; that is to say that repre-senting by F(x) the temperature of a point whose distance fromthe middle of the line is x, the initial temperature of the oppositepoint for which the distance is — x, has for value — F(x).

This second problem differs very little from the preceding, andmight be solved by a similar method: but the solution mayalso be derived from the analysis which has served to determinefor us the movement of heat in solids of finite dimensions.

Suppose that a part ab of the infinite prismatic bar h,as beenheated in any manner, see fig. (16*), and that the opposite partafi is in like state, but of contrary sign; all the rest of the solidhaving the initial temperature 0. Suppose also that the surround-

1 Of. Eiemann, Part. Biff. Gleich. § 16, p. 34. [A. F.]

SECT. I.] HEATED FINITE BAR. 341

ing-medium is maintained at .the constant temperature 0, and thatit receives heat from the bar or communicates heat to it through

Fig. 16*.

the external surface. It is required to find, after a given time frwhat will be the temperature v of a point whose distance from theorigin is x.

We shall consider first the heated bar as having a finitelength 2X, and as being submitted to some external cause whichmaintains its two ends at the constant temperature 0; we shallthen make X = oo«

352. We first employ the equation

dt'UJJdx2 CDSV; 0T dt~ arf~ '

and making v=e~ntn we have

du __ , cPudt " A? '

the general value of u may be expressed as. follows:

u = a^-^t sin ce + a2e~*v*H sing2x + a8e~*^ sin^o? + &c.

Making then x = X, which ought to make the value of vnothing, we have, to determine the series of exponents g, thecondition sin gX= 0, or gX=iir, i being an integer.

Hence

u1 = axe sin -y + a2e * sin —^ + &c.

It remains only to find the series of constants al9 a2, a3, &c.Making ( = 0we have

JP/ v . 7T . 27T^J . 37ra? .

u = F(x) = ax s i n j r + a2 sin -y - + a3 sin -w- + &c.


Let -rr = ^ and denote F.{x) or F(—J b y / ( r ) ; we have

f (r) = ax sin r + a2 sin 2r + a3 sin 3r + &c.

2 f

Now, we have previously found a* = - \drf{r) sinir, the inte-

gral being taken from r = 0 to r = w. Hence— at = jda)F(x)sm -^ .

The integral with respect to x must be taken from x = 0 to# = X. Making these substitutions, we form the equation

2 -us. f ~"*3M • 1TX [X 1 r, , . .

v = -^e"w A s m - y dxF (%) si-A- { - L J 0

7TOJ

sin -=

353. Such would be the solution if the prism had a finitelength represented by 2X. It is an evident consequence of theprinciples which we have laid down up to this point; it remainsonly to suppose the dimension X infinite. Let X— nir> n beingan infinite number; also let q be a variable whose infinitely small

increments dq are all equal; we write -y- instead of n. The general

term of the series which enters into equation (a) being

- * ^ 5 ^ . ITTX f . iirx

e * sin ~Y~ \ dx F (x) sm - y ,

we represent by %- the number i, which is variable and becomes

infinite. Thus we have

«2 ^2 ^2

Making these substitutions in the term in question we find

e-kqH s i n gX \cix ]? (^ s i n ^ gach of these terms must be divided

nrby X OYj-y becoming thereby an infinitely small quantity, and

SECT. I.] GENERAL SOLUTION. 343

the sum of the series is simply an integral, which must be takenwith respect to q from q = 0 to q = ao. Hence

v = - e~~M jdqe~k^H sinqx\dxF(x) sinqx (a),

the integral with respect to x must be taken from x = 0 to x = oo.We may also write

Q- JcqH g i n IJQ

orTTV

= e"w I c?a F(a) I dq e~kqH sin qx sin ja .Jo Jo

Equation (a) contains the general solution of the problem;and, substituting for F(x) any function whatever, subject or notto a continuous law, we shall always be able to express the valueof the temperature in terms of x and t: only it must be remarkedthat the function F(x) corresponds to a line formed of two equaland alternate parts1.

354. If the initial heat is distributed in the prism ia such amanner that the line FFFF (fig. 17), which represents the initial

Fig. 17.

state, is formed of two equal arcs situated right and left ofthe fixed point 0, the variable movement of the heat is expressedby the equation

Try Z*0 0 f00

= e-M \ fa]? (^ j dy e-kqH CQS yX CQS ^ x

Fig. 18.

f/£-

If the line ffff (fig. 18), which represents the initial state, is

i That is to say, F(x) = - F( - x). [A. F.]

formed of two similar and alternate arcs, the integral which givesthe value of the temperature is

= e~ht I cfo/(a) dq e~A'^sin qx sin qa.£ Jo Jo

If we suppose the initial heat to be distributed in any manner,it will be easy to derive the expression for v from the two preced-ing solutions. In fact, whatever the function </> (x) may be, whichrepresents the given initial temperature, it can always be decom-posed into two others F [x) + / (# ) , one of which corresponds to theline FFFF, and the other to the line ffff, so that we have thesethree conditions

F{x) = F{~ x),f(x) = - / ( - x), <f> (x) = F(x) +f(x).

We have already made use of this remark in Articles 233 and234. We know also that each initial state gives rise to a variablepartial state which is formed as if it alone existed. The composi-tion of these different states introduces no change into the tem-peratures which would have occurred separately from each ofthem. It follows from this that denoting by v the variable tem-perature produced by the initial state which represents the totalfunction <j> (x), we must have

-i /fee /*oo

— =ze"ht( I dq e-kq2tQOS qx I d%F(a) cos qx4 \J 0 JO

4- I dq e~h(^H sin qx I da f[p) sin qa 1.Jo Jo *

If we took the integrals with respect to a between the limits- Qo and + op, it is evident that we should double the results.We may then, in the preceding equation, omit from the firstmember the denominator 2, and take the integrals with respect toa in the second form a = — co toa = + oo. We easily see also

r+°o r+oothat we could write I da </> (a) cos ga, instead of I do. F{a) cos qa;

J - Q O J - oo

for it follows from the condition to which the function/(a) is sub-ject, that we must have

r+oo0 = I daf(oi) cos ff*.

J - C O

SECT. I.] ANY INITIAL DISTRIBUTION. 34s5

We can also write

I dx $ (a) sin qx instead of dxf(a) cos qx,J — oo J — oo

for we evidently have

/

+O0

dx F (a) sin qx.- 0 0

We conclude from thisToo / r + oo

7rv = e~ht\ dq e~h<lH doL<j> (a) cos ga cos qx•J Q \ J - oo

+ I rfa <£ (a) sin ^a sin ja? J ,J -00 /

Too /»+oo

or, 7rv = e~u\ dqe~kqHl dot. > (a) cos q (x — a),JO */ -oo

r+oo Too

or, 7rv = e~ht\ dx<j>(a) I dq e~kq2t cos q [x — a).J - oo JO

355. The solution of this second problem indicates clearlywhat the relation is between the definite integrals which we havejust employed, and the results of the analysis which we haveapplied to solids of a definite form. When, in the convergentseries which this analysis furnishes, we give to the quantitieswhich denote the dimensions infinite values; each of theterms becomes infinitely small, and the sum of the series isnothing but an integral. We might pass directly in the samemanner and without any physical considerations from the differenttrigonometrical series which we have employed in Chapter in. todefinite integrals; it will be sufficient to give some examples ofthese transformations in which the results are remarkable.

356. In the equation

•7 7r = sin u+ T: sin 3u + -=, sin 5u + &c.4 D O

we shall write instead of u the quantity - ; x is a new variable,

and n is an infinite number equal to -j- ; q is a quantity formed by

the successive addition of infinitely small parts equal to dq. We

shall represent the variable number i by —- . If in the general

term ^ — - sin (2/+ 1) -we put for i and n their values, the term

becomes -^sm2qx. Hence the sum of the series is £ —sm2qx,2q * *J q

the integral being taken from q = 0 to q = oo ; we have thereforethe equation \ TT = ^ I — sin 2qx which is always true whateverbe the positive value of x. Let 2qx = r, r being a new varia-ble, we have — = — and A TT = — sin r ; this value of the defi-

2 r 2 Jo r— sin r has been known for some time. If onr

supposing r negative we took the same integral from r = 0 tor = — oo, we should evidently have a result of contrary sign — \ TT.

357. The remark which we have just made on the value offdr

the integral — sin r, which is A TT or — A TT, serves to make known° J r

the nature of the expression

2 [°°dq sin q— -co&qx,

7? J 0 2

whose value we have already found (Article 348) to be equal to1 or 0 according as x is or is not included between 1 and — 1.

We have in fact

I — cos qx sin q — \ I -2 sin q (x + 1) — ^ I -2 sin q (x — 1) ;

the first term is equal to \ TT or — I TT according as x + 1 is a

positive or negative quantity; the second J I — sin q(x — 1) is equal

to \ TT or — J 7r, according as a? — 1 is a positive or negative quantity.Hence the whole integral is nothing if x + 1 and# — 1 have thesame sign; for, in this case, the two terms cancel each other. Butif these quantities are of different sign, that is to say if we have atthe same time

x + 1 > 0 and x - 1 < 0,

SECT. I.] PROPERTIES OF DEFINITE INTEGRALS. 347

the two terms add together and the value of the integral is | IT.

Hence the definite integral2 f °° do- —TTJO 2

sin q cos qx is a function of x

equal to 1 if the variable x has any value included between 1 and- 1; and the same function is nothing for every other value of xnot included between the limits 1 and — 1.

358. We might deduce also from the transformation of seriesinto integrals the properties of the two expressions2

2 Z1® dq cos qx , 2 f® qdq sin qx

Jo I T F ' a n d * Jo I T / - ;

the first (Art. 350) is equivalent to e~x when x is positive, and toex when # is negative. The second is equivalent to e~x if x is positive,and to — ex if # is negative, so that the two integrals have thesame value, when x is positive, and have values of contrary signwhen x is negative. One is represented by the line eeee (fig, 19),the other by the line eeee (fig. 20).

Fig. 19. Fig. 20.

The equation

1 . irx sin a sin x sin 2a sin 2x sin 3a sin 3x+ +

which we have arrived at (Art. 226), gives immediately the integral2 ["dq sin qir sin qx , . , . 3 . . , .- I —^—z^—2— j wnicn expression is equivalent to sm#, if a?is included between 0 and 7r, and its value is 0 whenever x ex-ceeds 7T.

1 At the limiting values of x the value of this integral is \ ; Riemann, § 15.2 Cf. Riemann, § 16.3 The substitutions required in the equation are — for #, dq for - , a for i—.

7T 7T 7T

We then have sin x equal to a series equivalent to the above integral for values of xbetween 0 and TT, the original equation being true for values of x between 0 and a.

[A.F.]

359. The same transformation applies to the general equation

^ 7r <f> (u) = sin u jdu <£ (u) sin u -f- sin 2u jdu (u) or </> I- j by f(x), and introduce into

the analysis a quantity q which receives infinitely small incre-

ments equal to dq, n will be equal to -j- and i to -^-; substituting

these values in the general term. ix [dx , fx\ . ix

sin — I — A [ - sin — ,n J n \nj n

we find dq sin qx jdxf{x) sin^x. The integral with respect to u

is taken from u = 0 to %i = 7r, hence the integration with respect tox must be taken from x = 0 to x = W7r, or from sc nothing to a;infinite.

We thus obtain a general result expressed by the equationToo Too

i irffa) — dq sin j# I cfoy (x) sin ja? (e),Jo Jo

for which reason, denoting by Q a function of q such that we have

f(u)= idqQsmqu an equation in which f(u) is a given function,

we shall have Q = - \duf(u) s i n ^ , the integral being taken from

u nothing to u infinite. We have already solved a similar problem(Art. 346) and proved the general equation

Too Too

\ IT F (x) = I dq cos qx j dxF {x) cos qx (e),Jo JO

which is analogous to the preceding. #

360. To give an application of these theorems, let us suppose/(as) -xr

y the second member of equation (e) by this substitution

becomes jdq sin qx jdx sin qx xr.

The integral

\dx sin ja? a?r or -Wl jqdx sin j ^ (qx)r

SECT. I.] CERTAIN DEFINITE INTEGRALS. 349

is equivalent to -T+i\dusintiurJ the integral being taken from u

nothing to u infinite.

Let a be the integral

du sin u ur;o

it remains to form the integral

\dq sin qx - ^ fi, or fjixr jdu sin u u~

denoting the last integral by v, taken from u nothing to u infinite,we have as the result of two successive integrations the termxr ixv. We must then have, according to the condition expressedby the equation (e),

J IT Xr = fJLV Xr Or fJLV = \ 7T J

t h u s t he product of the two t ranscendants

f °° f °° du\ duur sinu and I — u~r sin u is A7r.

Jo Jo u

For example, if r = — ~ , we find t h e known result

r°° d?w s in 7A / 7 r , x

—7=—= v 9 (fl);Vo v ^ v

in the same manner we findf cZw cos M

and from these two equations we might also conclude the following1,

dqe~q2 = g ^ /^ which has been employed for some time.

361. By means of the equations (e) and (e) we may solve thefollowing problem, which belongs also to partial differentialanalysis. What function Q of the variable q must be placed under

1 The way is simply to use the expressions e~z= + cos J-lz+ *J-lsin^/^lz,

transforming a and b by writing y2 for u and recollecting that v sj -1 = —-.-— .

Cf. § 407. [R. I . E.]

the integral sign in order that the expression IdqQe~^v may be

equal to a given function, the integral being taken from q nothingto q infinite1? But without stopping for different consequences,the examination of which would remove us from our chief object,we shall limit ourselves to the following result, which is obtainedby combining the two equations (e) and (e).

They may be put under the form

~7rf(x)= dq sin qx dxf (a) sin qx,£ Jo Jo

1 /.«> r°°and ^ irF (w) = I dq cos qx diF (a) cos qz.

* Jo Jo

If we took the integrals with respect to a from — oo to + oo,the result of each integration would be doubled, which is a neces-sary consequence of the two conditions

We have therefore the two equations

irf(cc) = I dq sin qx I daf(a) sin qx,J 0 J -oo

, 0 0 OO

and 7rF (X) = dq cos qx I daF(o) cos qx.JO J-oo

We have remarked previously that any function <£ (x) can.always be decomposed into two others, one of which F(x) satisfiesthe condition F (x) = F (— x), and the other f(x) satisfies thecondition/(#) =— /(— x). We have thus the two equations

/»+00 /«+OO

0 = / diF (a) sin qx, and 0 = I dxf(a) cos qz,J — oo J — oo

1 To do this write ±xj - 1 in f(x) and add, therefore

2 fq cos qx dq =/ (x J~l) +f(-x V3!))

which remains the same on writing - x for x,

therefore Q = - fdx [f(pcj~-l) + / ( -x j^ l . ) ] cos qx dx.

Again we may subtract and use the sine but the difficulty of dealing withimaginary quantities recurs continually. [R. L. E.]

SECT. I.] FOURIER'S THEOREM. 351

whence we conclude/.OO /.-J-OG

TT [F(x) +/(#)] = 7r</> [oo) = dq sin g# I dtf(a) sin jaJO J -00

/.OO /»+ 00

+ I dq cos jo? I diF (a) cos ja,JO J -oo

/.OO / . + 0O

and 7T > (a?) = I dfj s'm Qx \ dxj> (a) sin qaJ0 J -oo

-oo /.+ oo

4- d«2 cos qx I cfa<£ (a) cos qx,Jo J -°°

or 7r^(a?) = | dx(j>(a).l dq (smqxsinqx + cosqxcosqx);J -oo J o

- j - + oo -oo

or lastly1, <f> (x) = - I d^0 (a) dq cos q (x — a) [E).TTJ-OO JO

The integration with respect to q gives a function of x anda, and the second integration makes the variable a disappear.

Thus the function represented by the definite integral I dqcosq(x—a)

has the singular property, that if we multiply it by any function(f> (a) and by day and integrate it with respect to a between infinitelimits, the result is equal to irfy (%); so that the effect of the inte-gration is to change a into x, and to multiply by the number TT.

362. We might deduce equation (E) directly from the theorem1 Poisson,in his Memoire sur la Theorie des Ondes, in the Memoires de VAcademie

Jies Sciences, Tome i., Paris, 1818, pp. 85—87, first gave a direct proof of the theorem\ 00 +00

f{x) = - f dq f dae~k9 COS (qx-qa)f (a),TTJO J-<°

in which k is supposed to be a small positive quantity which is made equal to 0after the integrations.

Boole, On the Analysis of Discontinuous Functions, in the Transactions of theBoyal Irish Academy, Vol. xxi., Dublin, 1848, pp. 126—130, introduces some ana-lytical representations of discontinuity, and regards Fourier's Theorem as unprovedunless equivalent to the above proposition.

Deflers, at the end of a Note sur quelqites integrales definies &c.\ in the Bulletindes Sciences, Societe Philomatique, Paris, 1819, pp. 161—166, indicates a proof ofFourier's Theorem, which Poisson repeats in a modified form in the Journal Poly-technique, Cahier 19, p. 454. The special difficulties of this proof have beennoticed by De Morgan, Differential and Integral Calculus, pp. 619, 628.

An excellent discussion of the class of proofs here alluded to is given byMr J. W. L. Glaisher in an article On sin oo and cos oo , Messenger of Mathematics,Ser. i., Vol. v., pp. 232—244, Cambridge, 1871. [A. F.]

stated in Article 234, which gives the development of any func-tion F (cc) in a series of sines and cosines of multiple arcs. Wepass from the last proposition to those which we have just demon-strated, by giving an infinite value to the dimensions. Each termof the series becomes in this case a differential quantity1. Trans-formations of functions into trigonometrical series are some of theelements of the analytical theory of heat; it is indispensable tomake use of them to solve the problems which depend on thistheory.

The reduction of arbitrary functions into definite integrals,such as are expressed by equation (JE1), and the two elementaryequations from which it is derived, give rise to different conse-quences which are omitted here since they have a less direct rela-tion with the physical problem. We shall only remark that thesame equations present themselves sometimes in analysis underother forms. We obtain for example this result

<j)(x)=- dx<f)(a)l dq cos q(x- a)TTJ Q Jo

which differs from equation (JE) in that the limits taken withrespect to a are 0 and oo instead of being — oo and + oo .

In this case it must be remarked that the two equations (E)and (E1) give equal values for the second member when thevariable x is positive. If this variable is negative, equation (E')always gives a nul value for the second member. The same isnot the case with equation (E)} whose second member is equiva-lent to 7T(f> (x), whether we give to cc a positive or negative value.As to equation (E) it solves the following problem. To find afunction of x such that if x is positive, the value of the functionmay be <£ (x), and if x is negative the value of the function maybe always nothing2.

363. The problem of the propagation of heat in an infiniteline may besides be solved by giving to the integral of the partialdifferential equation a different form which we shall indicate in

1 Riemann, Part. Biff. Gleich. § 32, gives the proof, and deduces the formulaecorresponding to the cases F (x) = =t F ( - x),

2 These remarks are essential to clearness of view. The equations from which(E) and its cognate form may be derived will be found in Todhunter's IntegralCalculus, Cambridge, 1862, § 316, Equations (3) and (4). [A. F.]

SECT. I.] VARYING TEMPERATURE IN INFINITE BAR. 353

the following article. We shall first examine the case in whichthe source of heat is constant.

Suppose that, the initial heat being distributed in any mannerthroughout the infinite bar, we maintain the section A at aconstant temperature, whilst part of the heat communicated is dis-persed through the external surface. It is required to determinethe state of the prism after a given time, which is the object of thesecond problem that we have proposed to ourselves. Denoting by1 the constant temperature of the end A, by 0 that of the medium,

we have e K^ as the expression of the final temperature of apoint situated at the distance x from this extremity, or simply

TTTe~x, assuming for simplicity the quantity j ^ to be equal to unity.Denoting by v the variable temperature of the same point afterthe time t has elapsed, we have, to determine v, the equation

dt~CDdx2 GD8V}

let now v = e KSu,

, du _ K d\i HL ,we nave 'dt~CB'd^~ Cfi~Sa ;

du' _ , dV ,o r 7 — w 7 o "-^»

a^ decjr TTT

replacing -^ by k and -7—7 by k

T/, , , _^/ , du 7 d2w ,, , „ ,If we make u = e te we have -^- = /c -=-9; the value of w or

r — e KS is tha t of the difference between the actual and thefinal temperatures ; this difference u} which tends more and moreto vanish, and whose final value is nothing, is equivalent a t first to

denoting by F (x) the initial temperature of a point situated at thedistance x. Let / (a?) be the excess of the initial temperature over

F. H. 23

354 THEOEY OF HEAT. [CHAP. IX.

the final temperature, we must find for u a function which satisfies/¥'1 / ft *? /

the equation -y- = k-7-5 — hu, and whose initial value is / (# ) , andctV doo

IHL

final value 0. At the point A, or x = 0, the quantity v - e K®has, by hypothesis, a constant value equal to 0. We see by thisthat u represents an exgess of heat which is at first accumulated inthe prism, and which then escapes, either by being propagated toinfinity, or by being scattered into the medium. Thus to representthe effect which results from the uniform heating of the end A ofa line infinitely prolonged, we must imagine, 1st, that the line isalso prolonged to the left of the point A, and that each pointsituated to the right is now affected with the initial excess oftemperature; 2nd, that the other half of the line to the left ofthe point A is in a contrary state; so that a point situated at thedistance — x from the point A has the initial temperature —f(cc) :the heat then begins -to move freely through the interior of thebar, and to be scattered at the surface.

The point A preserves the temperature 0, and all the otherpoints arrive insensibly at the same state. In this manner we areable to refer the case in which the external source incessantly com-municates new heat, to that in which the primitive heat is propa-gated through the interior of the solid. We might therefore solvethe proposed problem in the same manner as that of the diffusionof heat, Articles 347 and 353; but in order to multiply methods ofsolution in a matter thus new, we shall employ the integral undera different form from that which we have considered up to thispoint.

364. The equation ^ = h -^ is satisfied by supposing u equal

to ex eu. This function of x and t may also be put under the formof a definite integral, which is very easily deduced from the known

value of Jdqe~q\ We have in fact Jw =Jdqe~q\ when the integral

is taken from y = - o o t o g = +oo. We have therefore also

v TT= jdqe'

SECT. I.J SOLUTION OF THE LINEAR EQUATION. 355

b being any constant whatever and the limits of the integral thesame as before. From the equation

J — 00

we conclude, by making b2 = kt

hence the preceding value of u or e"x ekt is equivalent to

we might also suppose u equal to the function

« f "*<**"*,

a and n being any two constants; and we should find in the sameway that this function is equivalent to

We can therefore in general take as the value of u the sum of aninfinite number of such values, and we shall have

'q2 (-******^ + ^= J+ &c.)

The constants a^ a2, a3, &c, and 7i1? w2,7i3, &c. being undetermined,the series represents any function whatever of x + 2qjkt; we have

therefore u = Jdqe'^(j>(a) + 2qjkt). The integral should be taken

from ^ = —oo to ^ = +oo, and the value of u will necessarily satisfydu 7 d2U rrn • • , i i » i

the equation -^- = k -y^. lnis integral which contains one arbi-trary function was not known when we had undertaken our re-searches on the theory of heat, which were transmitted to theInstitute of France in the month of December, 1807: it has been

23—2

given by M. Laplace1, in a work which forms part of volume VIII

of the M&noires de l'Ecole Polytechnique; we apply it simply tothe determination of the linear movement of heat. From it weconclude

v = e-h* fdqe-z2^ (x + 2qjkt) - <f * ^ ,J

when t = 0 the value of u is F(x) — e~x*™ orf(x);hence

1e-*2 <f>(ai) and <£ (x) = "]=f(x)mf(x) =

J - c

Thus the arbitrary function which enters into the integral, is deter-mined by means of the given function/(a?), and we have thefollowing equation, which contains the solution of the problem,

v im e~u f+c0

v = - e v ^ + -7=- dqer*f (x +

it is easy to represent this solution by a construction.

365. Let us apply the previous solution to the case in whichall points of the line AB having the initial temperature 0, the endA is heated so as to be maintained continually at the tempera-ture 1. It follows from this that F (x) has a nul value when x

differs from 0. Thus/(#) is equal to —e s whenever x differsfrom 0, and to 0 when x is nothing. On the other hand it isnecessary that on making x negative, the value oif(x) should changesign, so that we have the condition /(— x) = — f (x). We thusknow the nature of the discontinuous function/(#); it becomes

x JHL

— e "Vi2S when x exceeds 0, and +e ** when x is less than 0.We must now write instead of x the quantity x + 2q\/kt. To find

u or I d2e~q-f^f{x+ 2q\fkt)l we must first take the integral

from

x + 2q\/Wt = 0 to x + 2qslki- oo ,1 Journal de VEcole Polytechnique, Tome vnr. pp. 235—244, Paris, 1809.

Laplace shews also that the complete integral of the equation contains only onearbitrary function, but in this respect he had been anticipated by Poisson. [A. F.]

SECT. I.] APPLICATION OF THE SOLUTION. 357

and then from

cc + !q4ld = - oo to x + 2q*Ikt = 0.

For the first part, we have

and replacing h by its value - ^ we have

W7T

or - — i

orHLI™ HL

_V7T

7'/ TITf

Denoting the quantity 2 + y ^ ^ by r the preceding expressionbecomes

vM[dre->*,NTT J

this integral \dre-r* must be taken by hypothesis from

= 0 to

„ xor from q = — — r = to ^ = oo,

/ ^^OD

I it-Jut xor from r = K/ Q-QO — 7 ^ T to r = co.

The second part of the integral is


o r —j=-V7T

1 r JEk EM. fo r -1= e* ™eCDS \dre-r2,

VTT J

denoting by r the quantity q- A/jTfTa- The integral \dre'r

must be taken by hypothesis from

or from

from

The two last limits may, from the nature of the function e~^, bereplaced by these:

= — 00

= — 00

to q — — —

2

to r = — A

V C5

that is

»

2 / ^

r = s /5X« . x

It follows from this that the value of u is expressed thus:

fffi BU r 2 /si Hit r

u= e *KS eCJ)S dre~ — e***™eCI)S ldi"e-r\

the first integral must be taken from

HLt x

y CD

and the second from

SECT. I.] FORM OF SOLUTION IN CASE CONSIDERED. 359

Let us represent now the integral —j= ldre~r2 from r = R to r = ooSJTTJ

by yjr (B), and we shall have

HLt , x

HLthence u', which is equivalent to e CDS 9 is expressed by

and

v = e~*'s'Ts-e

The function denoted by -^ (i?) has been known for some time,and we can easily calculate, either by means of convergent series,or by continued fractions, the values which this function receives,when we substitute for M given quantities; thus the numericalapplication of the solution is subject to no difficulty1.

1 The following references are given by Riemann:Kramp. Analyse des refractions astronomiques et terrestres. Leipsic and Paris,

An. vn. 4to. Table I. at the end contains the values of the integral / e~P

from k=0-00 to 7c = 3 '00.Legendre. Traite des fonctions elliptiques et des integrates Euleriennes. Tomen.

350 THEOHY OF HEAT. [CHAP. IX.

366. If H be made nothing, we have

This equation represents the propagation of heat in an infinitebar, all points of which were first at temperature 0, except those atthe extremity which is maintained at the constant temperature 1.We suppose that heat cannot escape through the external surfaceof the bar; or, which is the same thing, that the thickness of thebar is infinitely great. This value of v indicates therefore the lawaccording to which heat is propagated in a solid, terminated byan infinite plane, supposing that this infinitely thick wall has firstat all parts a constant initial temperature 0, and that the surface issubmitted to a constant temperature 1. It will not be quiteuseless to point out several results of this solution.

Denoting by <£ (R) the integral —= jdr e~rZ taken from r = 0 to

r = ft, we have, when R is a positive quantity,

f (R) = !-</> (R) and ^ ( - R) =

hence

and v = l -

developing the integral $ (E) we have

Paris, 1826. 4to. pp. 520—1. Table of the values of the integral fdx (log-V*.

The first part for values of Hog - j from 0-00 to 0-50 j the second part for values

of x from 0-80 to 0*00.

Encke. Astronomisches Jahrbuchfilr 1834. Berlin, 1832, 8vo. Table I. at the2 /"*

end gives the values of -._ I e~*2dt from t^O-00 to * = 2'00. [A F.1. */w Jo

SECT. I.J MOVEMENT ACKOSS INFINITE PLANES. 361

hence, \ 5

1st, if we suppose x nothing, we find v = 1; 2nd, if x notbeing nothing, we suppose t = 0, the sum of the terms which

contain x represents the integral ldre~r2 taken from r = 0 to r = oo,

and consequently is equal to ^JTT; therefore v is nothing; 3rd,different points of the solid situated at different depths xv x2, x3,&c. arrive at the same temperature after different times tv t2, t3,&c. which are proportional to the squares of the lengths xx> x2, x3,&c; 4th, in order to compare the quantities of heat which duringan infinitely small instant cross a section S situated in the interiorof the solid at a distance x from the heated plane, we must take

(11)

the value of the quantity — KS -5- and we have

2K8

771tJ-rrt

dvthus the expression of the quantity -=- is entirely disengaged from

dx

the integral sign. The preceding value at the surface of the

heated solid becomes S ,— , which shews how the flow of heat

at the surface varies with the quantities C} D9 K, t; to find howmuch heat the source communicates to the solid during the lapseof the time t, we must take the integral

thus the heat acquired increases proportionally to the square root ofthe time elapsed.

367. By a similar analysis we may treat the problem of thediffusion of heat, which also depends on the integration of the

dv d2v

equation -j~ = k -^ — hv. Kepresenting by f{x) the initial tem-perature of a point in the line situated at a distance x from theorigin, we proceed to determine what ought to be the temperatureof the same point after a time t. Making v = e~htz, we havedz ffz r+co

-r==kjT> a n d consequently z = I dq e"^$ (x + 2q Jkt). Whent = 0, we must have

?2 <£ {x) or <j& (x) — ~]=-f (x) 5

hence

v = "7=" / cfo e-3 2 / (a; + 2^ v kt).

To apply this general expression to the case in which a part ofthe line from a? = - a t o # = a i s uniformly heated, all the rest ofthe solid being at* the temperature 0, we must consider that thefactor/(x-\-2qJkt) which multiplies e~q2 has, according to hypo-thesis, a constant value 1, when the quantity which is under thesign of the function is included between — a and a, and that allthe other values of this factor are nothing. Hence the integral

jdq e'*2 ought to be taken from x + 2q JFt = -atox + 2q Jlct = a,

or from q= -^~ij=- to <j = . . Denoting as above by -— yfr (R)—JTT

the integral \dre"^ taken from r = R to r = oo , we have

SECT. I.] COOLING OF AN INFINITE BAR. 363

368. We shall next apply the general equation

e-u /.+« _v = 1=] d(i e-qV(v + Zqjkt)

to the case in which the infinite bar, heated by a source ofconstant intensity 1, has arrived at fixed temperatures and isthen cooling freely in a medium maintained at the temperature0. For this purpose it is sufficient to remark that the initial

function denoted by f(x) is equivalent to e *k so long as thevariable x which is under the sign of the function is positive,

and that the same function is equivalent to eXsJ * when thevariable which is affected by the symbol/is less than 0. Hence

g— l i t . I* r^ /•

v = -j=( dqe-{*2e-*s/*e-2^hi+ \

the first integral must be taken from

oo + 2qJkt = O to oc+

and the second from

x + 2qjkt = - oo to x + 2qjkt = 0.

The first part of the value of v is

• / •

or

making r = q + Jht. The integral should be taken from

a = —j=. to q = oo,

or from r = Jht —r to r = oo .2^<


The second part of the value of v is

or

making r = q- JKt. The integral should be taken from

r — — oo to r = - Jht -^= ,JL \f rCt

or from r = Jht -\ j=^ to r = co,

whence we conclude the following expression:

369. We have obtained (Art. 367) the equation

to express the law of diffusion of heat in a bar of small thickness,heated uniformly at its middle point between the given limits

x = — a, x = + or.

We had previously solved the same problem by following adifferent method3 and we had arrived, on supposing a = 1, atthe equation

v = - e'M\ -^cos ## sin qe"^kt} (Art. 348).

To compare these two results we shall suppose in each x = 0 ;

denoting again by ^ ( 5 ) the integral jdr6"rZ taken from r = 0

to r = 22, we have

2e'ht

or * = 7 n

SECT. I.] IDENTITY OF DIFFERENT SOLUTIONS. 365

on the other hand we ought to have

7T J

- e [dqe*(1 - £ + ^7T J \ [O |D

or v = - e~M [dqe-***(1 - £ + ^ - &c.) .

Now the integral \dueru<it u*m taken from u = 0 to w = oo has

a known value, m being any positive integer. We have ingeneral

2 . 2 . 2 . 2 . . . 2

The preceding equation gives then, on making q*kt = u2,

«• 1 , «* 1 ,

or1 1 / i y I i/ I y i

2JFt) I2 5\2jki) ~ J 'Jir L2 Jkt 1 3 \2 JkV [2 •

This equation is the same as the preceding when we supposea = 1. We see by this that integrals which we have obtainedby different processes, lead to the same convergent series, andwe arrive thus at two identical results, whatever be the valueof x.

We might, in this problem as in the preceding, compare thequantities of heat which, in a given instant, cross differentsections of the heated prism, and the general expression of thesequantities contains no sign of integration; but passing by theseremarks, we shall terminate this section by the comparison ofthe different forms which we have given to the integral of theequation which represents the diffusion of heat in an infiniteline.

o ^ m ,. « ,, du 7 d2u

3/0. l o satisfy the equation -^i — ^-r~2i w e mSbJ assumeCut CLOG

u = e~x ekt, or in general u = e"nX en2kt, whence we deduce easily(Art. 364) the integral

u = I dqe-9' (}>{x+2q Jkt).

1 Cf. Kieniann, g 18.


From the known equation

we conclude

JTT^I dqe~(q+a)2, a being any constant; we have thereforeJ -oo

e°>2 =—=r \dq e'*2 e ' ^ , or

This equation holds whatever be the value of a. We may de-velope the first member; and by comparison of the terms we shall

obtain the already known values of the integral \dqe~q* qn. This

value is nothing when n is odd, and we find when n is an evennumber 2m,

% o* **. 1 . 3 . 5 . 7 . . . (2m -1) ,_

371. We have employed previously as the integral of thedu j d2u ,,

equation -^ = ^J7i ^ e expression

u = ag~n?kt cos nxx + a2e~n*ktcos w2# + a3e'n^ktcosw3# + &c.;

or this,

u = a£~n?u s in n ^ + a2e~"n^kt s in ??2^ + a3e~n*2Jct sin n3x + &c.

ax, a2, a8, &c, and w^ n2, n3, &c, being two series of arbitraryconstants. It is easy to see that each of these expressions isequivalent to the integral

jdq e~q2 sin n (x + 2q *Jkt), or jdq e~~q2 cos n(x+2q */B).

In fact, to determine the value of the integral

I dq e"q2 sin (x + 2q sfkt);

SECT. I.] IDENTITY OF SOLUTIONS. 367

we shall give it the following form

Idq erq% sin x cos 2q *Jkt + jdq e'q2 cos x sin 2q *Jkt;

or else,

J-oo \ 2 2 /

which is equivalent to

/ r ._ r ._e~** sin a; f i /dq e~{q' v"^)2 + i jdq e~(q+As/~'

cos x

the integral I dq e~{q± ^~kt)2 taken from q = ~ oo to £ = oo is VTT,

we have therefore for the value of the integral jdqe~q2 sin (x+2q \fkt)>

the quantity -vV e~** sin x> and in general

V-7T e'nm sin 7i = J dq e~q2 sin n (a? + 2q */lct),

we could determine in the same manner the integral

I dq e~* cos n [x + 2q *Jkt),y-oo

the value of which is VV e~n2^ cos n^.

We see by this that the integral

a8 sin n3cc + 68 cos nBx) + &c.

is equivalent to

1 r ra1sin«1(^ + 2 jV lVTTJ ^ t*! cos (x + 2# V/W) + b2 cos n2 (a? + 2q s/Tt) + &c.J '


The value of the series represents, as we have seen previously,any function whatever of x + 2q \[U; hence the general integralcan be expressed thus

The integral of the equation - ^ = k-^ may besides be pre-

sented under diverse other forms1. All these expressions are

necessarily identical.

SECTION II.

Of the free movement of heat in an infinite solid.

372. The integral of the equation -^ = -^f\ :r~2 (a) furnishesCut/ O J L J CLOG

immediately that of the equation with four variables

dv__ K (d2v d2v d2v\W + df + dzV

as we have already remarked in treating the question of the pro-pagation of heat in a solid cube. For which reason it is suflicientin general to consider the effect of the diffusion in the linearcase. "When the dimensions of bodies are not infinite, the distri-bution of heat is continually disturbed by the passage from thesolid medium to the elastic medium; or, to employ the expres-sions proper to analysis, the function which determines thetemperature must not only satisfy the partial differential equa-tion and the initial state, but is further subjected to conditionswhich depend on the form of the surface. In this case the integralhas a form more difficult to ascertain, and we must examine theproblem with very much more care in order to pass from the caseof one linear co-ordinate to that of three orthogonal co-ordinates :but when the solid mass is not interrupted, no accidental conditionopposes itself to the free diffusion of heat. Its movement is thesame in all directions.

1 See an article by Sir W. Thomson, " On the Linear Motion of Heat," Part I,Camb. Math. Journal, Vol. in. pp. 170—174. [A. F.]

SECT. II.] LINEAR MOVEMENT. 369

The variable temperature v of a point of an infinite line isexpressed by the equation

1 f+0

a? denotes the distance between a fixed point 0, and the point m,whose temperature is equal to v after the lapse of a time t Wesuppose that the heat cannot be dissipated through the externalsurface of the infinite bar, and that the initial state of the bar isexpressed by the equation v=f(x). The differential equation;which the value of v must satisfy, is

dv __ K d'v , vW dx~2 W '

But to simplify the investigation, we write

dv d2v

which assumes that we employ instead of t another unknown

i + K t

equal to ^ .

If i n / (x), a function of a? and constants, we substitute x + 2n s/1

for x, and if, after having multiplied by —~ e~n2, we integrate with

respect to n between infinite limits, the expression

satisfies, as we have proved above, the differential equation (b);that is to say the expression has the property of giving the samevalue for the second fluxion with respect to x, and for the firstfluxion with respect to t From this it is evident that a functionof three variables f(x>y,z) will enjoy a like property, if we substi-tute for x, y, z the quantities

x + 2nji, y 4- 2pjt, z + 2%Jt,

provided we integrate after having multiplied by

F. H. 24

In fact, the function which we thus form,

gives three terms for the fluxion with respect to t, and these threeterms are those which would be found by taking the second fluxionwith respect to each of the three variables x, y7 z.

Hence the equation

t, = 7r-5 jdn [dpjdqe-(n2+*>2+q2)f(x + 2njT, y + 2pjt,z

gives a value of v which satisfies the partial differential equation

dv d2v d2v d2v . p .

373. Suppose now that a formless solid mass (that is to sayone which fills infinite space) contains a quantity of heat whoseactual distribution is known. Let v=F(x, y, z) be the equationwhich expresses this initial and arbitrary state, so that themolecule whose co-ordinates are x, y, z has an initial temperatureequal to the value of the given function F(x,y,z). We canimagine that the initial heat is contained in a certain part ofthe mass whose first state is given by means of the equationv=-F(xf y, z), and that all other points have a nul initial tem-perature.

It is required to ascertain what the system of temperatureswill be after a given time. The variable temperature v mustconsequently be expressed by a function <£ (x, yf z, t) which oughtto satisfy the general equation (A) and the condition <f> (x, y, z, 0)= F[x, y, z). Now the value of this function is given by theintegral

v = ir'* J dn J dp J dq <r<»a+pa+«8> F (x + 2n Jt, y + 2p JI, z + 2q JT).

In fact, this function v satisfies the equation (A), and if in it wemake t = 0, we find

7r~e I dnj dpi dq e-(n2+2>2+<z2) F (^ y} ^

or, effecting the integrations, F (x, yf z).

SECT. II.] THE CASE OF THREE DIMENSIONS. 371

374. Since the function v or <f> (x, y9 z, t) represents theinitial state when in it we make t = 0, and since it satisfies thedifferential equation of the propagation of heat, it represents alsothat state of the solid which exists at the commencement of thesecond instant, and making the second state vary, we concludethat the same, function represents the third state of the solid, andall the subsequent states. Thus the value of v> which we havejust determined, containing an entirely arbitrary function of threevariables x, y, z, gives the solution of the problem; and we cannotsuppose that there is a more general expression, although other-wise the same integral may be put under very different forms.

Instead of employing the equation

we might give another form to the integral of the equation

-ji^-j-** a n d it would always be easy to deduce from it the

integral which belongs to the case of three dimensions. Theresult which we should obtain would necessarily be the same asthe preceding.

To give an example of this investigation we shall make use ofthe particular value which has aided us in forming the exponentialintegral.

Taking then the equation -=- = ~rj ... (&), let us give to v the

very simple value e~nH cosnx, which evidently satisfies the

differential equation (b). In fact, we derive from it -y- = — n2vCut

and -j-r2 = — n2v. Hence also, the integralax

r dn e~nH cos nx

belongs to the equation (6) ; for this value of v is formed of thesum of an infinity of particular values. Now, the integral

rdne~nH cos nx

24—2


is known, and is known to be equivalent to / - /- (see the follow-

ing article). Hence this last function of x and t agrees also with

the differential equation (b). I t is besides very easy to verify

directly that the particular value —jf satisfies the equation in

question.

The same result will occur if we replace the variable x byx — a, a being any constant. We may then employ as a particular

Ae ~~value the function ^ — , in which we assign to a any value

f A e—4T~whatever. Consequently the sum I dzf (a) p — also satisfies

the differential equation (b); for this sum is composed of aninfinity of particular values of the same form, multiplied byarbitrary constants. Hence we can take as a value of v in the

,. dv d2v ,1 - «equation -^- = -—a the following,

Ae ~

A being a constant coefficient. If in the last integral we supposeX A? = 22, making also A = —— , we shall have

« •

We see by this how the employment of the particular values

e-nH c o s noc o r __^

leads to the integral under a finite form.

SECT. II.] EVALUATION OF AN INTEGRAL. 373

375. The relation in which these two particular values are toeach other is discovered when we evaluate the integral1

- 0 0

dne~nH cos nx.

To effect the integration, we might develope the factor cos nxand integrate with respect to n. We thus obtain a series whichrepresents a known development; but the result may be derived

more easily from the following analysis. The integral I dn e~nHco& nx

is transformed to I dp e"^2cos 2pu7 by assuming nH =^2and nx = 2pu.

We thus have

I dn e~nH cos nx = -T.l dpe~#2 cos 2pu.

We shall now write

/dp e~p2cos 2pu = \ jdpe~^+2^uy/-ri + J idp e-^-

tp-v^V2 +1 e-u2 idp e-fc+W^)1.

Now each of the integrals which enter into these two terms isequal to Jir. We have in fact in general

r+oo

*JTT = I dq e~Q2,

and consequently

whatever be the constant b. We find then on making

b = + uJ~^T, jdq e~q2 cos 2qu = e~u* JTT,

hence I dne~nH cos nx = . ^ ,

1 The value is obtained by a different method in Todhunter's Integral Calculus,§375. [A.F.]


and putting for u its value —-p, we have2 Vt

cos w# =

ft 4*t

Moreover the particular value —j=- is simple enough to present

itself directly without its being necessary to deduce it from thevalue e~nH cos nx. However it may be, it is certain that the

X2

function —p- satisfies the differential equation -j- = -j— - it is theJt dt dx

_same consequently with the function —j=—, whatever the quan-

tity a may be.

376. To pass to the case of three dimensions, it is sufficient( ) 2

to multiply the function of x and t, — j=- , by two other similar

functions, one of y and t, the other of z and t; the product willevidently satisfy the equation

dvd2v d2v d?v

We shall take then for v the value thus expressed:

v = t 2 e

If now we multiply the second member by da, d/3, dy, and byany function whatever/ (a, /3, 7) of the quantities a, /3, 7, we find,on indicating the integration, a value of v formed of the sum of aninfinity of particular values multiplied by arbitrary constants.

It follows from this that the function v may be thus ex-pressed :

r+0° r+°° r+v=j doi\ dft

J —00 J -"-GO J —0

This equation contains the general integral of the proposedequation (A): the process which has led us to this integral ought

SECT. II.] INTEGRAL FOR THREE DIMENSIONS. 375

to be remarked since it is applicable to a great variety of cases;it is useful chiefly when the integral must satisfy conditionsrelative to the surface. If we examine it attentively we perceivethat the transformations which it requires are all indicated bythe physical nature of the problem. We can also, in equation (j),change the variables. By taking

(a-x)2 2

we have, on multiplying the second member by a constant co-efficient A,

v = 23A jdnfdpfdq erW+&+f>f (x + 2n Jt, y + 2p*Ji, z + 2q Ji).

Taking the three integrals between the limits — oo and -f oo,and making t = 0 in order to ascertain the initial state, we find

3

v = 2z Air~^f{xiy9 z). Thus, if we represent the known initialtemperatures by F (x9 y, z), and give to the constant A the value

- 3 - ?

2 IT 2, we arrive at the integralf+Q0

dn—00

\ dpJ — 00 J - CO

which is the same as that of Article 372.

The integral of equation (A) may be put under several otherforms, from which that is to be chosen which suits best theproblem which it is proposed to solve.

It must be observed in general, in these researches, that twofunctions <£ (%, y, z, t) are the same when they each satisfy thedifferential equation (A), and when they are equal for a definitevalue of the time. It follows from this principle that integrals,which are reduced, when in them we make t = 0, to the samearbitrary function F(x9 y, z), all have the same degree of generality;they are necessarily identical.

The second member of the differential equation (a) was

multiplied by ^ , and in equation (b) we supposed this coefficient

equal to unity. To restore this quantity, it is sufficient to write


777. instead of t, in the integral (j) or in the integral (J). We

shall now indicate some of the results which follow from theseequations.

377. The function which serves as the exponent of thenumber e* can only represent an absolute number, which followsfrom the general principles of analysis, as we have proved ex-plicitly in Chapter II., section IX. If in this exponent we replace

jr.

the unknown t by ^yr, we see that the dimensions of K, C,D and t,

with reference to unit of length, being — 1, 0, — 3, and 0, the

dimension of the denominator - ^ is 2 the same as that of each

term of the numerator, so that the whole dimension of the expo-nent is 0. Let us consider the case in which the value of t increasesmore and more; and to simplify this examination let us employfirst the equation

(a-*)2

which represents the diffusion of heat in an infinite line. Supposethe initial heat to be contained in a given portion of the line,from x = — htox = -\-ff, and that we assign to x a definite value X>which fixes the position of a certain point m of that line. If the

2 1 JY

time t increase without limit, the terms -r— and —-.— which' U At

enter into the exponent will become smaller and smaller absolutea?8 _2aaJ a 2

numbers, so that in the product e~~~*t e ±t e~& we can omitthe two last factors which sensibly coincide with unity. We thusfind

This is the expression of the variable state of the line after avery long time; it applies to all parts of the line which are lessdistant from the origin than the point m. The definite integral

xl* In such quantities as e~ **. [A. F.]

SECT. II.] INITIAL HEAT COLLECTED AT THE ORIGIN. 377

foL) denotes the whole quantity of heat B contained in the—h

solid, and we see that the primitive distribution has no influenceon the temperatures after a very long time. They depend onlyon the sum By and not on the law according to which the heat hasbeen distributed.

378. If we suppose a single element co situated at the originto have received the initial temperature/, and that all the othershad initially the temperature 0, the product cof will be equal to

r+gthe integral I doif(a) or B. The constant/is exceedingly great

J —h

since we suppose the line co very small.

The equation v = —j=—j= cof represents the movement which2 *j7T*Jt

would take place, if a single element situated at the origin hadbeen heated. In fact, if we give to x any value a, not infinitely

x^

small, the function —- will be nothing when we suppose t = 0.

The same would not be the case if the value of x were

nothing. In this case the function —— receives on the contraryJt

an infinite value when t = 0. We can ascertain distinctly thenature of this function, if we apply the general principles of thetheory of curved surfaces to the surface whose equation is

The equation v = —,_ ., cof expresses then the variable tem-2 sJlT sjt

perature at any point of the prism, when we suppose the wholeinitial heat collected into a single element situated at the origin.This hypothesis, although special, belongs to a general problem,since after a sufficiently long time, the variable state of the solid isalways the same as if the initial heat had been collected at theorigin. The law according to which the heat was distributed, has


much influence on the variable temperatures of the prism; butthis effect becomes weaker and weaker, and ends with being quiteinsensible.

379. I t is necessary to remark that the reduced equation (y)does not apply to that part of the line which lies beyond the pointm whose distance has been denoted by X.

In fact, however great the value of the time may be, we might

choose a value of x such that the term e u would differ sensiblyfrom unity, so that this factor could not then be suppressed. Wemust therefore imagine that we have marked on either side of theorigin 0 two points, m and m, situated at a certain distance X or— X, and that we increase more and more the value of the time,observing the successive states of the part of the line which isincluded between m and m. These variable states converge moreand more towards that which is expressed by the equation

Whatever be the value assigned to X, we shall always be able tofind a value of the time so great that the state of the line momdoes not differ sensibly from that which the preceding equation (y)expresses.

If we require that the same equation should apply to otherparts more distant from the origin, it will be necessary to supposea value of the time greater than the preceding.

The equation (y) which expresses in all cases the final state ofany line, shews that after an exceedingly long time, the differentpoints acquire temperatures almost equal, and that the temperaturesof the same point end by varying in inverse ratio of the squareroot of the times elapsed since the commencement of the diffusion.The decrements of the temperature of any point whatever alwaysbecome proportional to the increments of the time.

380. If we made use of the integral

J 2jirkt

SECT. II.] ADMISSIBLE SIMPLIFICATIONS. 379

to ascertain the variable state of the points of the line situated ata great distance from the heated portion, and in order to express

the ultimate condition suppressed also the factor e 4Jct , theresults which we should obtain would not be exact. In fact,supposing that the heated portion extends only from a = 0 to a — gand that the limit g is very small with respect to the distance x ofthe point whose temperature we wish to determine; the quantity

which forms the exponent reduces in fact to — JJ- ; that,,

(a — xf a?is to say the ratio of the two quantities ., and JJ- approaches

more nearly to unity as the value of x becomes greater withrespect to that of a: but it does not follow that we can replaceone of these quantities by the other in the exponent of e. Ingeneral the omission of the subordinate terms cannot thus takeplace in exponential or trigonometrical expressions. The quanti-ties arranged under the symbols of sine or cosine, or under theexponential symbol ey are always absolute numbers, and we canomit only the parts of those numbers whose value is extremelysmall; their relative values are here of no importance. To decideif we may reduce the expression

fgdaf(a) i~^ to im PW(a),Jo Jo

we must not examine whether the ratio of x to a is very great,

but whether the terms ry- , -JJ- are very small numbers. This

condition always exists when t the time elapsed is extremely great;

but it does not depend on the ratio - .

381. Suppose now that we wish to ascertain how much timeought to elapse in order that the temperatures of the part of thesolid included between x = 0 and x = X, may be represented verynearly by the reduced equation


and that 0 and g may be the limits of the portion originallyheated.

The exact solution is given by the equation

2jirkt

and the approximate solution is given by the equation

(y).

Kk denoting the value 7 ^ of the conducibility. In order that theequation (y) may be substituted for the preceding equation (i), it

is in general requisite that the factor e *w , which is that whichwe omit, should differ very little from unity; for if it were 1 or \we might apprehend an error equal to the value calculated or to

the half of that value. Let then e 4M =1 + CO, CO being a small

fraction, as r-™ or 7^7/, from this we derive the condition

4ikt co \

and if the greatest value g which the variable a can receive is1 qx

very small with respect to x, we have t = - ^r •CO £ru

We see by this result that the more distant from the originthe points are whose temperatures we wish to determine by meansof the reduced equation, the more necessary it is for the value ofthe time elapsed to be great. Thus the heat tends more and moreto be distributed according to a law independent of the primitiveheating. After a certain time, the diffusion is sensibly effected,that is to say the state of the solid depends on nothing more thanthe quantity of the initial heat, and not on the distribution whichwas made of it. The temperatures of points sufficiently near tothe origin are soon represented without error by the reducedequation (y)\ but it is not the same with points very distant from

SECT. II.] NUMERICAL APPLICATION. 381

the source. We can then make use of that equation only whenthe time elapsed is extremely long. Numerical applications makethis remark more perceptible.

382. Suppose that the substance of which the prism is formedis iron, and that the portion of the solid which has been heated isa decimetre in length, so that g = 0 1 . If we wish to ascertainwhat will be, after a given time, the temperature of a point mwhose distance from the origin is a metre, and if we employ forthis investigation the approximate integral (y), we shall commitan error greater as the value of the time is smaller. This errorwill be less than the hundredth part of the quantity sought, if thetime elapsed exceeds three days and a half.

In this case the distance included between the origin 0 and thepoint m, whose temperature we are determining, is only ten timesgreater than the portion heated. If this ratio is one hundredinstead of being ten, the reduced integral (y) will give the tem-perature nearly to less than one hundredth part, when the valueof the time elapsed exceeds one month. In order that the ap-proximation may be admissible, it is necessary in general, 1st that

the quantity —JJ-—• should be equal to but a very small fraction

as Too or iooo or *ess; 2nc*'ttat t^ie error w^ c^ must

should have an absolute value very much less than the smallquantities which we observe with the most sensitive thermometers.

When the points which we consider are very distant from theportion of the solid which was originally heated, the temperatureswhich it is required to determine are extremely small; thus theerror which we should commit in employing the reduced equationwould have a very small absolute value; but it does not followthat we should be authorized to make use of that equation. Forif the error committed, although very small, exceeds or is equal tothe quantity sought; or even if it is the half or the fourth, or anappreciable part, the approximation ought to be rejected. It isevident that in this case the approximate equation (y) would notexpress the state of the solid, and that we could not avail ourselvesof it to determine the ratios of the simultaneous temperatures oftwo or more points.

383. It follows from this examination that we ought not toj rg J±T*P

conclude from the integral v = —7== d*f(<z).e «« that the

law of the primitive distribution has no influence on the tempera-ture of points very distant from the origin. The resultant effectof this distribution soon ceases to have influence on the pointsnear to the heated portion; that is to say their temperaturedepends on nothing more than the quantity of the initial heat,and not on the distribution which was made of it: but greatnessof distance does not concur to efface the impress of the distribu-tion, it preserves it on the contrary during a very long timeand retards the diffusion of heat. Thus the equation

JIT J&JctJo

only after an immense time represents the temperatures of pointsextremely remote from the heated part. If we applied it withoutthis condition, we should find results double or triple of the trueresults, or even incomparably greater or smaller; and this wouldnot only occur for very small values of the time, but for greatvalues, such as an hour, a day, a year. Lastly this expressionwould be so much the less exact, all other things being equal, asthe points were more distant from the part originally heated.

384. When the diffusion of heat is effected in all directions,the state of the solid is represented as we have seen by theintegral

If the initial heat is contained in a definite portion of the solidmass, we know the limits which comprise this heated part, andthe quantities a, /?, 7, which vary under the integral sign, cannotreceive values which exceed those limits. Suppose then that wemark on the three axes six points whose distances are + X, +Y} -\-Z,and — X, — Yy — Z> and that we consider the successive states ofthe solid included within the six planes which cross the axes atthese distances; we see that the exponent of e under the sign of

SECT. II.] APPROXIMATE FORMULA. 383

integration, reduces to - (* +^jJ~Z)y w h e n t h e v a l u e o f t h e t i m e

, 2ax A a2

increases without limit. In fact, the terms such as ^ and ^

receive in this case very small absolute values, since the numera-tors are included between fixed limits, and the denominatorsincrease to infinity. Thus the factors which we omit differextremely little from unity. Hence the variable state of thesolid, after a great value of the time, is expressed by

The factor \di \dft \dyf(% /3, 7) represents the whole quantity

of heat B which the solid contains. Thus the system of tempera-tures depends not upon the initial distribution of heat, but onlyon its quantity. We might suppose that all the initial heat wascontained in a single prismatic element situated at the origin,whose extremely small orthogonal dimensions were co1, co2, o>3. Theinitial temperature of this element would be denoted by anexceedingly great number / , and all the other molecules of thesolid would have a nul initial temperature. The productco1co2w3f is equal in this case to the integral

Whatever be the initial heating, the state of the solid whichcorresponds to a very great value of the time, is the same as if allthe heat had been collected into a single element situated at theorigin.

385. Suppose now that we consider only the points of thesolid whose distance from the origin is very great with respectto the dimensions of the heated part; we might first imaginethat this condition is sufficient to reduce the exponent of e inthe general integral. The exponent is in fact

_ («-«,)»+ (ft-y^+fr-g)' _4>kt '


and the variables a, /?, 7 are, by hypothesis, included betweenfinite limits, so that their values are always extremely smallwith respect to the greater co-ordinate of a point very remotefrom the origin. It follows from this that the exponent of eis composed of two parts M+JJL, one. of which is very smallwith respect to the other. But from the fact that the ratio

-— is a very small fraction, we cannot conclude that the ex-ponential eM+* becomes equal to eM, or differs only from it bya quantity very small with respect to its actual value. We mustnot consider the relative values of M and /n, but only the absolutevalue of fjb. In order that we may be able to reduce the exactintegral (j) to the equation

£ *J IT fC It

it is necessary that the quantity

2dx + 2/3y + 2yz - a2 - /32 - 72

whose dimension is 0, should always be a very small number.If we suppose that the distance from the origin to the point m,whose temperature we wish to determine, is very great withrespect to the extent of the part which was at first heated,we should examine whether the preceding quantity is alwaysa very small fraction &>. This condition must be satisfied toenable us to employ the approximate integral

but this equation does not represent the variable state of thatpart of the mass which is very remote from the1 source of heat.It gives on the contrary a result so much the less exact, allother things being equal, as the points whose temperature weare determining are more distant from the source.

The initial heat contained in a definite portion of the solidmass penetrates successively the neighbouring parts, and spreadsitself in all directions; only an exceedingly small quantity ofit arrives at points whose distance from the origin is very great.

SECT. III.] HIGHEST TEMPERATURES IN A SOLID. 385

When we express analytically the temperature of these points,the object of the investigation is not to determine numericallythese temperatures, which are not measurable, but to ascertaintheir ratios. Now these quantities depend certainly on the lawaccording to which the initial heat has been distributed, and theeffect of this initial distribution lasts so much the longer as theparts of the prism are more distant from the source. But if the

2a22xX aterms which form part of the exponent, such as -77- and -rj-, have

absolute values decreasing without limit, we may employ theapproximate integrals.

This condition occurs in problems where it is proposed todetermine the highest temperatures of points very distant fromthe origin. We can demonstrate in fact that in this case thevalues of the times increase in a greater ratio than the distances,and are proportional to the squares of these distances, when thepoints we are considering are very remote from the origin. It isonly after having established this proposition that we can effectthe reduction under the exponent. Problems of this kind are theobject of the following section.

SECTION III.

Of the highest temperatures in an infinite solid.

386. We shall consider in the first place the linear move-ment in an infinite bar, a portion of which has been uniformlyheated, and we shall investigate the value of the time which mustelapse in order that a given point of the line may attain itshighest temperature.

Let us denote by 2g the extent of the part heated, the middleof which corresponds with the origin 0 of the distances x. All thepoints whose distance from the axis of?/ is less than# and greaterthan — gy have by hypothesis a common initial temperature / , andall other sections have the initial temperature 0. We supposethat no loss of heat occurs at the external surface of the prism, or,which is the same thing, we assign to the section perpendicular tothe axis infinite dimensions. It is required to ascertain what will

F. H. 9~>

be the time t which corresponds to the maximum of temperatureat a given point whose distance is x.

We have seen, in the preceding Articles, that the variabletemperature at any point is expressed by the equation

1 f (<x-*)2

v = —p==. \daf(a)e *** .

The coefficient k represents 7 T ^ , K being the specific con-O x /

ducibility, C the capacity for heat, and D the density.

To simplify the investigation, make k = l, and in the result

write let or -™ instead of t. The expression for v becomes

V =1 -0

This is the integral of the equation -=- = -r-2. The function -=-

measures the velocity with which the heat flows along the axis of

the prism. Now this value of -y-- is given in the actual problemctcc

without any integral sign. We have in factdv f [+°j OL-X J ^ li- = %dx —7— e «d 4tf

or, effecting the integration,

dx 2 Jirt {

d2v387. The function -=-s may also be expressed without the

sign of integration: now it is equal to a fluxion of the first order -^;

hence on equating to zero this value of -=- , which measures thectt

instantaneous increase of the temperature at any point, we havethe relation sought between x and t We thus find

SECT. III.] TIMES OF HIGHEST TEMPERATURES. 387

which gives

{x+g)e~

whence we conclude

We have supposed ^ = = 1. To restore the coefficient we

must write - ^ instead of t, and we have

. ffGD x

The highest temperatures follow each other according to thelaw expressed by this equation. If we suppose it to represent thevarying motion of a body which describes a straight line, x beingthe space passed over, and t the time elapsed, the velocity ofthe moving body will be that of the maximum of temperature.

When the quantity g is infinitely small, that is to say when theinitial heat is collected into a single element situated at the

origin, the value of t is reduced to --, and by differentiation or

development m series we rind - ^ = -^ .

We have left out of consideration the quantity of heat whichescapes at the surface of the prism; we now proceed to take accountof that loss, and we shall suppose the initial heat to be containedin a single element of the infinite prismatic bar.

388. In the preceding problem we have determined thevariable state of an infinite prism a definite portion of which wasaffected throughout with an initial temperature f. We supposethat the initial heat was distributed through a finite space fromx = 0 to x — b.

We now suppose that the same quantity of heat bf is containedin an infinitely small element, from x = 0 to x = co. The tempera-

25—2

ture of the heated layer will therefore b e — , and from this follows

what was said before, that the variable state of the solid isexpressed by the equation

Kthis result holds when the coefficient - ^ which enters into the

differential equation -y = -7™ -y-2 - hv, is denoted by k As to the^ dt CD dx

777coefficient h, it is equal to rrn^; S denoting the area of the

section of the prism, I the contour of that section, and H theconducibility of the external surface.

Substituting these values in the equation (a) we have

aCB

~Hl' (A);

f represents the mean initial temperature, that is to say, thatwhich a single point would have if the initial heat were distributedequally between the points of a portion of the bar whose lengthis I, or more simply, unit of measure. It is required to determinethe value t of the time elapsed, which corresponds to a maximumof temperature at a given point.

To solve this problem, it is sufficient to derive from equationa/1)

(a) the value of -5-, and equate it to zero; we have

dv , x2 lv , 1 2ifel Uk ...Jt=-hv + W<v-2t> a n d f-ltl-lF ®>

hence the value 6, of the time which must elapse in order that thepoint situated at the distance x may attain its highest temperature,is expressed by the equation

SECT. III.] VALUES OF HIGHEST TEMPERATUKES. 389

To ascertain the highest temperature F, we remark that the

exponent of e'1 in equation (a) is ^ + 7X/- Now equation (b)

= i t - \; hence u + m = i t - 5 *and putting for J i t s

#2 / i A

known value, we have ht + -rj- = \ / 7 + 7 #2 J substituting this ex-

ponent of e"1 in equation (a), we havey

Jkd '

and replacing / A; by its known value, we find, as the expressionof the maximum V,

The equations (c) and (d) contain the solution of the problem;SI K

let us replace h and k by their values 77™ and - ^ ; let us also

write ~ g instead of j , representing by g the semi-thickness of the

prism whose base is a square. We have to determine Fand 0,the equations

These equations are applicable to the movement of heat in athin bar, whose length is very great. We suppose the middle ofthis prism to have been affected by a certain quantity of heat hfwhich is propagated to the ends, and scattered through the convexsurface* F denotes the maximum of temperature for the pointwhose distance from the primitive source is x; 6 is the timewhich has elapsed since the beginning of the diffusion up to theinstant at which the highest temperature F occurs. The coeffi-


cients C, H, K, D denote the same specific properties as in thepreceding problems, and g is the half-side of the square formed bya section of the prism.

389. In order to make these results more intelligible by anumerical application, we may suppose that the substance of whichthe prism is formed is iron, and that the side 2g of the square isthe twenty-fifth part of a metre.

We measured formerly, by our experiments, the values of Hand K; those of C and D were already known. Taking the metreas the unit of length, and the sexagesimal minute as the unit oftime, and employing the approximate values of H, Ky C, D, weshall determine the values of V and 9 corresponding to a givendistance. For the examination of the results which we have in view,it is not necessary to know these coefficients with great precision.

We see at first that if the distance x is about a metre and a2H

half or two metres, the term -^- cc2, which enters under the radical,K9

has a large value with reference to the second term T . The ratio

of these terms increases as the distance increases.

Thus the law of the highest temperatures becomes more andmore simple, according as the heat removes from the origin. Todetermine the regular law which is established through the wholeextent of the bar, we must suppose the distance x to be verygreat, and we find

ora CDjg6= 3 . V ^- .T?

390. We see by the second equation that the time which corre-sponds to the maximum of temperature increases proportionallywith the distance. Thus the velocity of the wave (if however wemay apply this expression to the movement in question) is constant,or rather it more and more tends to become so, and preserves thisproperty in its movement to infinity from the origin of heat.

SECT. III.] LAW OF THE HIGHEST TEMPERATURES. 391

We may remark also in the first equation that the quantityi—

fe~~x*K~g expresses the permanent temperatures which thedifferent points of the bar would take, if we affected the originwith a fixed temperature / , as may be seen in Chapter L,Article 76.

In order to represent to ourselves the value of V, we musttherefore imagine that all the initial heat which the source con-tains is equally distributed through a portion of the bar whoselength is 5, or the unit of measure. The temperature/ whichwould result for each point of this portion, is in a manner themean temperature. If we supposed the layer situated at theorigin to be retained at a constant temperature/ during an infinitetime, all the layers would acquire fixed temperatures whose

general expression is fe Kg 9 denoting by x the distance of thelayer. These fixed temperatures represented by the ordinates ofa logarithmic curve are extremely small, when the distance isconsiderable; they decrease, as is known, very rapidly, accordingas we remove from the origin.

Now the equation (§) shews that these fixed temperatures,which are the highest each point can acquire, much exceed thehighest temperatures which follow each other during the diffusionof heat. To determine the latter maximum, we must calculatethe value of the fixed maximum, multiply it by the constant

number f =-J ~7r=> a n d divide by the square root of the dis-

tance x.

Thus the highest temperatures follow each other through thewhole extent of the line, as the ordinates of a logarithmic curvedivided by the square roots of the abscissae, and the movement ofthe wave is uniform. According to this general law the heatcollected at a single point is propagated in direction of the lengthof the solid.

391. If we regarded the conducibility of the external surfaceof the prism as nothing, or if the conducibility K or the thickness2g were supposed infinite, we should obtain very different results.


227We could then omit the term -=- x*} and we should have1

K9f

F = —p-7=

In this case the value of the maximum is inversely propor-tional to the distance. Thus the movement of the wave wouldnot be uniform. It must be remarked that this hypothesis ispurely theoretical, and if the conducibility H is not nothing, butonly an extremely small quantity, the velocity of the wave is notvariable in the parts of the prism which are very distant from theorigin. In fact, whatever be the value of H} if this value is given,as also those of K and g, and if we suppose that the distance x

2Hincreases without limit, the term -~- x2 will always become much

9

greater than J. The distances may at first be small enough for2H

the term -^^x2 to be omitted under the radical. The times areKg

then proportional to the squares of the distances; but as the heatflows in direction of the infinite length, the law of propagationalters, and the times become proportional to the distances. Theinitial law, that is to say, that which relates to points extremelynear to the source, differs very much from the final law which isestablished in the very distant parts, and up to infinity: but, inthe intermediate portions, the highest temperatures follow eachother according to a mixed law expressed by the two precedingequations (Z?) and (0),

392. It remains for us to determine the highest temperaturesfor the case in which heat is propagated to infinity in every direc-tion within the material solid. This investigation, in accordancewith the principles which we have established, presents nodifficulty.

When a definite portion of an infinite solid has been heated,and all other parts of the mass have the same initial temperature 0,heat is propagated in all directions, and after a certain time thestate of the solid is the same as if the heat had been originallycollected in a single point at the origin of co-ordinates. The time

1 See equations (D) and (C), article 388, making 6 - 1 . [A. F.]

SECT. III.] GENERAL INVESTIGATION. 393

which must elapse before this last effect is set up is exceedinglygreat when the points of the mass are very distant from the origin.Each of these points which had at first the temperature 0 isimperceptibly heated; its temperature then acquires the greatestvalue which it can receive; and it ends by diminishing more andmore, until there remains no sensible heat in the mass. Thevariable state is in general represented by the equation

v=jdafdbfdce~ g J ^ f(a, b,c) (E).

The integrals must be taken between the limits

a = -al9 a = a2> b = -b1, b = b2, c = -cv c = c2.

The limits -alf + a2, -blt + b2, -c19 +c2 are given; theyinclude the whole portion of the solid which was originally heated.The function f(a, b, c) is also given. It expresses the initialtemperature of a point whose co-ordinates are a, b, c. The defi-nite integrations make the variables a, b, c disappear, and thereremains for v a function of x, yy z> t and constants. To determinethe time 0 which corresponds to a maximum of v, at a given point

m, we must derive from the preceding equation the value of -^;(to

we thus form an equation which contains 6 and the co-ordinates ofthe point m. From this we can then deduce the value of 0. Ifthen we substitute this value of 6 instead of t in equation (E), wefind the value of the highest temperature V expressed in x, y, zand constants.

Instead of equation (E) let us write

v =jdajdbfdc Pf(a9 b, c),

denoting by P t h e multiplier of / (a , b, c), we have

dv 3 v

393. We must now apply the last expression to points of thesolid which are very distant from the origin. Any point what-ever of the portion which contains the initial heat, having for co-ordinates the variables a, b, c, and the co-ordinates of the point m

S94 THEORY OF HEAT. [CHAP. IX.

whose temperature we wish to determine being xy y, zy the square ofthe distance between these two points is (a — xf + (b — y)2+ (c — zf\

eft)and this quantity enters as a factor into the second term of -7-.

Now the point m being very distant from the origin, it isevident that the distance A from any point whatever of the heatedportion coincides with the distance D of the same point from theorigin; that is to say, as the point m removes farther and fartherfrom the primitive source, which contains the origin of co-ordinates,the final ratio of the distances D and A becomes 1.

It follows from this that in equation (e) which gives the valueft tj

of -r the factor (a — xf + (b — yf + (c - zf may be replaced bydt

x2 + y2 + z2 or r2, denoting by r the distance of the point m fromthe origin. We have then

dvor m'- vw ~ 2tj •

If we put for v its value, and replace t by -pf.y i n order to

re-establish the coefficient ^ which we had supposed equal to 1,

we have

dv f r2

dt 4^V 2^ 7r'

394. This result belongs only to the points of the solid whosedistance from the origin is very great with respect to the greatestdimension of the source. It must always be carefully noticed thatit does not follow from this condition that we can omit the varia-bles a, 6, c under the exponential symbol. They ought only to beomitted outside this symbol. In fact, the term which enters underthe signs of integration, and which multiplies / ( a , b, c), is the

SECT. III.] CONDITIONS FOR DISTANT POINTS. 395

product of several factors, such as-gi lax -x2

Kt Kt A Kt

Now it is not sufficient for the ratio - to be always a verya

great number in order that we may suppress the two first factors.If, for example, we suppose a equal to a decimetre, and x equal toten metres, and if the substance in which the heat is propagated isiron, we see that after nine or ten hours have elapsed, the factor

"laxKt

e CD is still greater than 2 ; hence by suppressing it we should/•//it

reduce the result sought to half its value. Thus the value of ~Y ,as it belongs to points very distant from the origin, and for anytime whatever, ought to be expressed by equation (a). But it isnot the same if we consider only extremely large values of thetime, which increase in proportion to the squares of the distances :in accordance with this condition we must omit, even under theexponential symbol, the terms which contain a, b, or c. Now thiscondition holds when we wish to determine the highest tempera-ture which a distant point can acquire, as we proceed to prove.

dv395. The value of -7- must in fact be nothing in the case in

(Jutquestion ; we have therefore

r2 3 A JT 1 20 '

2ZCD

Thus the time which must elapse in order that a very distantpoint may acquire its highest temperature is proportional to thesquare of the distance of this point from the origin.

If in the expression for v we replace the denominator — -

2by its value ^ r2, the exponent of e'1 which is


may be reduced to ~, since the factors which we omit coincide with

unity. Consequently we find

3v = -g Ida jdb tdcf(a, b, c).

The integral Ida jdb \dcf(a, b, c) represents the quantity of

the initial heat: the volume of the sphere whose radius is r is4~7rr3, so that denoting b y / t h e temperature which each molecule

of this sphere would receive, if we distributed amongst its parts

all the initial heat, we shall have v = A / —if

The results which we have developed in this chapter indicatethe law according to which the heat contained in a definite portionof an infinite solid progressively penetrates all the other partswhose initial temperature was nothing. This problem is solvedmore simply than that of the preceding Chapters, since byattributing to the solid infinite dimensions, we make the con-ditions relative to the surface disappear, and the chief difficultyconsists in the employment of those conditions. The generalresults of the movement of heat in a boundless solid mass arevery remarkable, since the movement is not disturbed by theobstacle of surfaces. It is accomplished freely by means of thenatural properties of heat. This investigation is, properlyspeaking, that of the irradiation of heat within the materialsolid.

SECTION IV.

Comparison of the integrals.

396. The integral of the equation of the propagation of heatpresents itself under different forms, which it is necessary to com-pare. It is easy, as we have seen in the second section of thisChapter, Articles 372 and 376, to refer the case of three dimen-sions to that of the linear movement; it is sufficient therefore tointegrate the equation

dv K d2v

SECT. IV.] FORM OF THE INTEGRAL FOR A RING. 397

or the equation

iv_=<?l (a).dt dx2

To deduce from this differential equation the laws of the propa-gation of heat in a body of definite form, in a ring for example,it was necessary to know the integral, and to obtain it under acertain form suitable to the problem, a condition which could befulfilled by no other form. This integral was given for the firsttime in our Memoir sent to the Institute of France on the21st of December, 1807 (page 124, Art. 84): it consists in thefollowing equation, which expresses the variable system of tem-peratures of a solid ring :

R is the radius of the mean circumference of the ring ; the integralwith respect to a must be taken from a = 0 to a = 2TTR, or, whichgives the same result, from a = - ITR to a = 7TJR ; i is any integer,and the sum 2 must be taken from i = — oc to i = + oo ; t> denotesthe temperature which would be observed after the lapse of atime tf at each point of a section separated by the arc x from thatwhich is at the origin. We represent by v = F (x) the initial tem-perature at any point of the ring. We must give to i the succes-sive values

0, + 1 , +2, + 3 , &c, and - 1, - 2 , - 3 , &c,

t (i* ~ n iand instead of cos - —„—'- write

M

ix ia . ix . iaC0SEC0SB+SmMsmli-

We thus obtain all the terms of the value of v. Such is theform under which the integral of equation (a) must be placed, inorder to express the variable movement of heat in a ring (Chap, iv.,Art. 241). We consider the case in which the form and extent ofthe generating section of the ring are such, that the points of thesame section sustain temperatures sensibly equal. We supposealso that no loss of heat occurs at the surface of the rin«\


397. The equation (a) being applicable to all values of B> wecan suppose in it B infinite; in which case it gives the solution ofthe following problem. The initial state of a solid prism ofsmall thickness and of infinite length, being known and expressedby v = F(x), to determine all the subsequent states. Consider theradius B to contain numerically n times the unit radius of thetrigonometrical tables. Denoting by q a variable which successivelybecomes dq, 2dq} 3dq,...idq, &c, the infinite number n may

-t

be expressed by -r~, and the variable number i by -^-. Making

these substitutions we find

v= —%dq I dzF(a) e~qH cos q (x — a).

The terms which enter under the sign 2 are differential quan-tities, so that the sign becomes that of a definite integral; andwe have

1 f+0° r+0°v = x—I do. Jfr (a) I dq e~qt cos (qx — qoc) (73).

^ 7 r J - o o J-ao N2 * J x^;

This equation is a second form of the integral of the equation(a); it expresses the linear movement of heat in a prism of infinitelength (Chap. VII., Art. 354). It is an evident consequence of thefirst integral (a).

398. We can in equation (£) effect the definite integrationwith respect to q; for we have, according to a known lemma, whichwe have already proved (Art. 375),

IJ - 0

dz e~*2cos 2hz = e~h2J^rr.00

Making then z2 = q2t, we find

dqe~q2t cos (qx - qa) = ^ 5 / W .-°° Jt

Hence the integral (/3) of the preceding Article becomes

(7)-

SECT. IV.J LAPLACE'S FORM OF THE INTEGRAL. 399

If we employ instead of a another unknown quantity /?,

making — = fi, we find

i r

(«).

This form (8) of the integral1 of equation (a) was given inVolume Vlii. of the Memoir es de VEcole Poly technique, by M.Laplace,who arrived at this result by considering the infinite series whichrepresents the integral.

Each of the equations (/3), (7), (8) expresses the linear diffusionof heat in a prism of infinite length. It is evident that these arethree forms of the same integral, and that not one can be con-sidered more general than the others. Each of them is containedin the integral (a) from which it is derived, by giving to R aninfinite value.

399. It is easy to develope the value of v deduced fromequation (a) in series arranged according to the increasing powersof one or other variable. These developments are self-evident,and we might dispense with referring to them; but they give riseto remarks useful in the investigation of integrals. Denoting by

(f>'} <£", <f>", &c, the functions -7-<£(#), -7-2 <£(#), 7-3 0 0*0 > &c-> w e

have

dvv = c+ldtv ;

1 A direct proof of the equivalence of the forms

1 /•+» _ 2 _ i ^7 = 1 dpe * <f>(x + 2p>Jt) and e d^fl<p{x)i (see Art. 401)

sjirj-co n

has been given by Mr Glaisher in the Messenger of Mathematics, June 1876, p. 30.Expanding <f>(x+2p*Jt) by Taylor's Theorem, integrate each term separately:terms involving uneven powers of *Jt vanish, and we have the second form;which is therefore equivalent to

1 roo /*oo

- / da I dq e~qHCOS q(a-x) 0(a),w y-00 Jo

from which the first form may be derived as above. We have thus a slightlygeneralized form of Fourier's Theorem, p. 351. [A. F.]


here the constant represents any function of x. Putting for v" its

value c" + ldtviy, and continuing always similar substitutions, we

find

v = c + fdtv"

= c + jdt \c" +fdt (clv +fdt v«J\ ,

or v = c + tc"+^J* + ^c«+?Lc«ii + &cL2 12 L?

In this series, c denotes an arbitrary function of x. If we wishto arrange the development of the value of v, according to ascend-ing powers of x, we employ

d2v _ dvd%2~dt'

and, denoting by (f>/} (/>//? cf>J/jy &c. the functions

d . d* d*

Jt+> 3 ? ^ 5 ? * &a?

we have first v = a + bx+ idx \dx vj; a and b here represent any

two functions of t We can then put for v its value

a, + &/e + jdx \dx vn ;

and for vit its value au + b^x + jdxjdxv///} and so on. By continued

substitutions

= a

\dxvt

jdx (ad + hx -h |da? frfarv,)

jdxjdz at + & ar + /ku jku ^a^ + 6/y.r + f .r f^ V/\\

SECT. IV.] NUMBER OF ARBITRARY FUNCTIONS. 401

X X X . n

o r v •• ' '

^ t ^btt + &c (X).

In this series, a and b denote two arbitrary functions of t

If in the series given by equation (X) we put, instead ofa and b, two functions </> (t) and -^ (f), and develope them accordingto ascending powers of t, we find only a single arbitrary functionof oot instead of two functions a and b. We owe this remark toM. Poispon, who has given it in Volume VI. of the Mdmoires deTEcole Poly technique page 110.

Reciprocally, if in the series expressed by equation (T) we de-velope the function c according to powers of x, arranging theresult with respect to the same powers of x, the coefficients ofthese powers are formed of two entirely arbitrary functions of t;which can be easily verified on making the investigation.

400. The value of v9 developed according to powers of t,ought in fact to contain only one arbitrary function of x; for thedifferential equation (a) shews clearly that, if we knew, as afunction of x, the value of v which corresponds to t = 0, theother values of the function v which correspond to subsequentvalues of t, would be determined by this value.

It is no less evident that the function v, when developedaccording to ascending powers of x, ought to contain two com-pletely arbitrary functions of the variable t. In fact the differential

equation -^—9 = -y- shews that, if we knew as a function of t the1 da? dtvalue of v which corresponds to a definite value of a?,1 we couldnot conclude from it the values of v which correspond to all theother values of x. I t would be necessary in addition, to give asa function of t the value of v which corresponds to a second valueof x, for example, to that which is infinitely near to the first. Allthe other states of the function v, that is to say those which corre-spond to all the other values of x, would then be determined. Thedifferential equation (a) belongs to a curved surface, the verticalordinate of any point being vf and the two horizontal co-ordinates

F. H. 26

x and and t It follows evidently from this equation (a) that theform of the surface is determined, when we give the form of thevertical section in the plane which passes through the axis of x :and this follows also from the physical nature of the problem; forit is evident that, the initial state of the prism being given, all thesubsequent states are determined. But we could not constructthe surface, if it were only subject to passing through a curvetraced on the first vertical plane of t and v. It would be necessaryto know further the curve traced on a second vertical planeparallel to the first, to which it may be supposed extremely near.The same remarks apply to all partial differential equations, andwe see that the order of the equation does not determine in allcases the number of the arbitrary functions.

401. The series (T) of Article 399, which is derived from theequation

dv d2v , .w

may be put under the form v — etiyi ($> (x). Developing the ex-dl

ponential according to powers of D, and writing -y-. instead of D\

considering i as the order of the differentiation, we have

Following the same notation, the first part of the series (X)

(Art. 399), which contains only even powers of x, may be expressed

under the form cos {x y^TZ?) (/> (t). Develope according to powers

of x, and write -^ instead of D\ considering i as the order of the

differentiation. The second part of the series (X) can be derivedfrom the first by integrating with respect to x, and changing thefunction </> (t) into another arbitrary function -f (f). We havetherefore

W9

W= f fVd:vjQ ^ v j[ r \J

SECT. IV.] SYMBOLICAL METHODS. 403

This known abridged notation is derived from the analogywhich exists between integrals and powers. As to the use madeof it here, the object is to express series, and to verify themwithout any development. It is sufficient to differentiate underthe signs which the notation employs. For example, from theequation v=et2)2 <f) (x), we deduce, by differentiation with respectto t only,

which shews directly that the series satisfies the differentialequation (a). Similarly, if we consider the first part of the series(X), writing

v = cos (x J- D) $ (t),

we have, differentiating twice with respect to x only,

Hence this value of v satisfies the differential equation (a).

We should find in the same manner that the differentialequation

+ ° ( 6 )

gives as the expression for v in a series developed according toincreasing powers of y,

v = cos (yD) </> (x).

We must develope with respect to y, and write -j- instead ofax

D: from this value of v we deduce in fact,

The value sin (yD) ty (x) satisfies also the differential equation;hence the general value of v is

v = cos (yD) (j> (x) + W, where W= sin (yD) -vjr (x).

26—2

402. If the proposed differential equation is

<Fv d2v t d2v / v+ {C)

and if we wish to express v in a series arranged according topowers of t, we may denote by D</> the function

—- 6 — A-

and the equation being —2 = Dv, we haveat

v = cos (t J— D) <£ (a?, y).

From this we infer thatd\ __ -p. _ d2v d2v

We must develope the preceding value of v according to powers/ d2 d2\*

of t, write ( —2 + -^—2), instead of D% and then regard i as the order\cLz£ cLy /

of differentiation.

The following value jdt cos (tj— D) ty (x, y) satisfies the same

condition; thus the most general value of v is

Q = cos (t J— D) <j> (pc, y) + W;

and W= jdt cos (t J^HJ) ^ (x, y);

v is a function f(x, y> t) of three variables. If we make t = 0, we

h a v e / = {x, y,O) = <j> (x, y); and denoting ^ / (a? , y, i) b y / («, y,«),

we h a v e / (a?, y,0) = y]r [x, y).

If the proposed equation is

the value of v in a series arranged according to powers of t will

SECT. IV.] A DIFFERENTIAL EQUATION. 405

d2

be v = cos (tD2) $ fa y\ denoting ^—2 by D\ for we deduce fromthis value

d2v

The general value of v, which can contain only two arbitraryfunctions of x and y, is therefore

v = cos (tD2) <j> (x, y) + Wy

and W=f*dt cos (tD2) f te, y).Jo

Denoting v by f(x, y} t), and ^ by / ' (x, y, t), we have to

determine the two arbitrary functions,

$ 0 . y) =f(x> y> °)> and f (x> y) = / fa y, o).

403. If the proposed differential equation is

d2v d'v d*v dSdf^ dx^ dx'dy2^ dy*~~

we may denote by D(f> the function -j-%- + -~2> so that DD<f>

or D2cj> can be formed by raising the binomial (-r-2 + 3~2J to the\ctiOc cty /

second degree, and regarding the exponents as orders of differen-d2v

tiation. Equation (e) then becomes - r^+ i)2?; = 0; and the valuectt

of v, arranged according to powers of t, is cos (tD) <f> (x, y); forfrom this we derive

d*v_ d2v^d4v a d*v d*v ,df—Vv> o r d f + ^ + ^ + { )

The most general value of v being able to contain only twoarbitrary functions of x and y, which is an evident consequence ofthe form of the equation, may be expressed thus:

v = cos (tD) <p (x, y) + I dt cos (tD) ^ (#, y)t

The functions <f> and ^ are determined as follows, denoting the

function v hjffa y, t), and jf (x, y, t) b y / fa y, t)9

4> fa y) =f fa y> °)> f fa y) = / i fa v>Lastly, let the proposed differential equation be

dv d2v , , d4v d% , d8v .

the coefficients a, by c, d are known numbers, and the order of theequation is indefinite.

The most general value of v can only contain one arbitraryfunction of x; for it is evident, from the very form of the equa-tion, that if we knew, as a function of x, the value of v whichcorresponds to t = 0, all the other values of v, which correspond tosuccessive values of t, would be determined. To express v, weshould have therefore the equation v = etD(f> {x)\

We denote by I)cf> the expression

C + & C

that is to say, in order to form the value of v, we must developaccording to powers of t, the quantity

and then write ^ - instead of a, considering the powers of a as orders

of differentiation. In fact, this value of v being differentiatedwith respect to t only, we have

dv detD. . . d\ - d*v dev

It would be useless to multiply applications of the same process.For very simple equations we can dispense with abridged expres-sions ; but in general they supply the place of very complex in-vestigations. We have chosen, as examples, the preceding equa-tions, because they all relate to physical phenomena whose analyticalexpression is analogous to that of the movement of heat. The twofirst, (a) and (b), belong to the theory of heat; and the three

SECT. IV.J OTHER MODES OF INTEGRATION. 407

following (c), (d), (e), to dynamical problems; the last (/) ex-presses what the movement of heat would be in solid bodies, ifthe instantaneous transmission were not limited to an extremelysmall distance. We have an example of this kind of problem inthe movement of luminous heat which penetrates diaphanousmedia.

404. We can obtain by different means the integrals of theseequations: we shall indicate in the first place that which resultsfrom the use of the theorem enunciated in Art. 361, which wenow proceed to recal.

If we consider the expression

I doi cf> (a) I dcf> cos (px—poi), (a)J - 00 J - 0 0

we see that it represents a function of x\ for the two definiteintegrations with respect to a and p make these variables dis-appear, and a function of x remains. The nature of the functionwill evidently depend on that which we shall have chosen for<f) (a). We may ask what the function c/> (a), ought to be, in orderthat after two definite integrations we may obtain a given functionf{x). In general the investigation of the integrals suitable forthe expression of different physical phenomena, is reducible toproblems similar to the preceding. The object of these problemsis to determine the arbitrary functions under the signs of thedefinite integration, so that the result of this integration may bea given function. It is easy to see, for example, that the generalintegral of the equation

dv d2v 7 d4v dGv 7d

8v p+ h + + d + &

would be known if, in the preceding expression (a), we coulddetermine <j> (a), so that the result of the equation might be agiven function/ (x). In fact, we form directly a particular valueof v, expressed thus,

= e~mtcospx,

and we find this condition,

m = ajf + bp* + qf + &c.

We might then also take

v = e-mt c o s (pX —pz),

giving to the constant a any value. We have similarly

v=(dx<l>('j.) e-tW+W+^+Wcos {px -px).

It is evident that this value of v satisfies the differential equation( / ) ; it is merely the sum of particular values.

Further, supposing t = 0, we ought to find for v an arbitraryfunction of x. Denoting this function by/(a?), we have

f {x)>— \ d* <j> (OL) I dp cos ( px — pi).j J

Now it follows from the form of the equation (f), that the mostgeneral value of v can contain only one arbitrary function of x.In fact, this equation shews clearly that if we know as a functionof x the value of v for a given value of the time t, all the othervalues of v which correspond to other values of the time, arenecessarily determined. It follows rigorously that if we know,as a function of t and x, a value of v which satisfies the differentialequation; and if further, on making t = 0, this function of x and tbecomes an entirely arbitrary function of x, the function of x andt in question is the general integral of equation (/) . The wholeproblem is therefore reduced to determining, in the equationabove, the function (/> (a), so that the result of two integrationsmay be a given function f(x). It is only necessary, in order thatthe solution may be general, that we should be able to take forf(x). an entirely arbitrary and even discontinuous function. It ismerely required therefore to know the relation which must alwaysexist between the given function f(x) and the unknown function<j6 (a). Now this very simple relation is expressed by the theoremof which we are speaking. It consists in the fact that when theintegrals are taken between infinite limits, the function <j> (a) is

9 ~ / ( a ) 5 ^ a t is to say, that we have the equation

SECT. IV.] VIBKATION OF ELASTIC LAMINA. 409

From this we conclude as the general integral of the proposedequation ( / ) ,

v = ^ f dif(z) /+ dp e-tW+W+w'+Wcos(px-poi) ...(c).ZTTJ-OO J -oo

405. If we propose the equation

£+£-• «•which expresses the transverse vibratory movement of an elasticplate1, we must consider that, from the form of this equation, themost general value of v can contain only two arbitrary functionsof x: for, denoting this value "of v by f(x,t), and the function

-rf(®, t) by / ' (a?, t), it is evident that if we knew fix, 0) andat

dvf (x, 0), that is to say, the values of v and — at the first instant,

dtall the other values of v would be determined.

This follows also from the very nature of the phenomenon. Infact, consider a rectilinear elastic lamina in its state of rest: x isthe distance of any point of this plate from the origin of co-ordinates; the form of the lamina is very slightly changed, bydrawing it from its position of equilibrium, in which it coincidedwith the axis of x on the horizontal plane; it is then abandoned toits own forces excited by the change of form. The displacement issupposed to be arbitrary, but very small, and such that the initialform given to the lamina is that of a curve drawn on a verticalplane which passes through the axis of x. The system will suc-cessively change its form, and will continue to move in the verticalplane on one side or other of the line of equilibrium. The mostgeneral condition of this motion is expressed by the equation

d2v d4v A ,

df+c& = ° W'Any point m, situated in the position of equilibrium at a

distance x from the origin 0, and on the horizontal plane, has, at1 An investigation of the general equation for the lateral vibration of a thin

elastic rod, of which (d) is a particular case corresponding to no permanentinternal tension, the angular motions of a section of the rod being also neglected,will be found in Donkin's Acoustics, Chap. ix. §§ 169—177. [A.F.]

410 THEORY OF HEAT. ["CHAP. IX.

the end of the time t, been removed from its place through theperpendicular height v. This variable flight v is a function ofx and t The initial value of v is arbitrary; it is expressed by anyfunction </> (x). Now, the equation (d) deduced from the funda-mental principles of dynamics shews that the second fluxion

72

of v, taken with respect to t, or — , and the fluxion of the fourthctt

d4vorder taken with respect to x, or - ^ are two functions of x and t,

doc

which differ only in sign. We do not enter here into the specialquestion relative to the discontinuity of these functions; we havein view only the analytical expression of the integral.

"We may suppose also, that after having arbitrarily displacedthe different points of the lamina, we impress upon them verysmall initial velocities, in the vertical plane in which the vibrationsought to be accomplished. The initial velocity given to anypoint m has an arbitrary value. It is expressed by any function

x) of the distance so.

It is evident that if we have given the initial form of thesystem or c/> (x) and the initial impulses or A/T (#), all the subse-quent states of the system are determinate. Thus the functionv or f{x> t), which represents, after any time t} the correspondingform of the lamina, contains two arbitrary functions (j> (x)and yfr (x).

To determine the function sought f(x71), consider that in theequation

we can give to v the very simple value

u = cos q2t cos qx,

or else u = cos qH cos (qx — qa);

denoting by q and a any quantities which contain neither x nor t.We therefore also have

u = I dz F(a) jdq cos (ft cos {qx — qo),

SECT. IV.] SOLUTION OF EQUATION OF VIBRATION. 4 1 1

F(OL) being any function, whatever the limits of the integrationsmay be. This value of v is merely a sum of particular values.

Supposing now that t = 0, the value of v must necessarilybe that which we have denoted b y / ( # , 0) or </>(&•). We havetherefore

4>(oc) = \diF (a) \dq cos {qx — qa).

The function F{a) must be determined so that, when the twointegrations have been effected, the result shall be the arbitraryfunction </> (x). Now the theorem expressed by equation (B) shewsthat when the limits of both integrals are — oo and + GO , wehave

Hence the value of u is given by the following equation:I r+QO /•+»

u = — I da cf> (a) dq cos q2t cos (qx — qa).

If this value of u were integrated with respect to t, the </> init being changed to - r, it is evident that the integral (denotedby W) would again satisfy the proposed differential equation (d)>and we should have

W= — I da -^r (a) \dq—l sin qH cos {qx — qx).

This value W becomes nothing when £ = 0; and if we take theexpression

dWdt

r -I r+co -+00

= — da yjr (a) I dq cos (ft cos {qx — qa),^7T J -co J - 0 0

we see that on making t = 0 in it, it becomes equal to y{r (x).

The same is not the case with the expression -^-\ it becomesetc

nothing when t = 0, and u becomes equal to <£ {x) when t = 0.

It follows from this that the integral of equation (d) is0 = 5 - 1 da <j> (a) I dq cos q2t cos (ax - qa) + W= u+W,

47T J - c o J - o o

and

W = 5 - I rZx i/r (a) I dq -z sin 2^ cos {qx - qa)."K J -CO J - 0 0 *7

In fact, this value of v satisfies the differential equation (d);also when we make t = 0; it becomes equal to the entirely arbitrary

dvfunction <£ (x); and when we make t = 0 in the expression ^-,

it reduces to a second arbitrary function yfr (a?). Hence the valueof v is the complete integral of the proposed equation, and therecannot be a more general integral.

406. The value of v may be reduced to a simpler form byeffecting the integration with respect to q. This reduction, andthat of other expressions of the same kind, depends on the tworesults expressed by equations (1) and (2), which will be provedin the following Article.

J dq COS qH COS qz=—sm^ + ^J (1),

J ^sin^cos2^ = ^ s^ ^ ^ J (2).From this we conclude

Denoting -^ by another unknown JUL, we have£ sj t

a = x + 2/JL Ji, da = 2dfx Jt

fir \Putting in place of sin f ~ + [^) its value

we have

U= ITT-

We have proved in a special memoir that (S) or (8'), theintegrals of equation (d), represent clearly and completely themotion of the different parts of an infinite elastic lamina. Theycontain the distinct expression of the phenomenon, and readilyexplain all its laws. It is from this point of view chiefly that we

SECT. IV.] TWO DEFINITE INTEGRALS. 413

have proposed them to the attention of geometers. They shewhow oscillations are propagated and set up through the wholeextent of the lamina, and how the effect of the initial displace-ment, which is arbitrary and fortuitous, alters more and more asit recedes from the origin, soon becoming insensible, and leavingonly the existence of the action of forces proper to the system, theforces namely of elasticity.

407. The results expressed by equations (1) and (2) dependupon the definite integrals

I dx cos x2, and I dx sin x*;

r+oo r+oo

g = dx cos x2y and h = I dx sin re2;

J — oo J - oolet

and regard g and h as known numbers. It is evident that in thetwo preceding equations we may put y + b instead of x, denotingby b any constant whatever, and the limits of the integral will bethe same. Thus we have

g = P^ycos (y2 + 2by + 62), h= r°°*/sin (y2+ 2by+b2),J - CO J — 00

J y

cos y2 cos 2hy cos b2 — cos y2 sin 2by sin b- sin y2 sin 2by cos b2 — sin y2 cos 2by sin b

Now it is easy to see that all the integrals which contain thefactor sin 2by are nothing, if the limits are — oo and + oo ; forsin 2by changes sign at the same time as y. We have therefore

g = cosb2 I dy cosy*cos2by - sin b2 j dy siny2cos 2by (a).

The equation in h also gives

j | \ f sin y2 cos 2by cos b2 + cos t/2 cos 2by sin b2) t

J \ + cos y2 sin 2by cos b2 — sin y2 sin 2by sin &2} '

and, omitting also the terms which contain sin 2byy we have

^ = cos b2 j dy sin y2 cos 2£y + sin b2 j dy cosy2 cos 2by (6).

THEORY OF HEAT. [CHAP. IX.

The two equations (a) and (b) give therefore for g and h thetwo integrals

jdy sin y2 cos 2by and jdy cos ?/2 cos 2by,

which we shall denote respectively by A and B. We may nowmake

y2=p\ &nd2by = pz; or y=p\/t, b = —j=:2\lt

we have therefore

cos pH cos£>£ = J., Vtf jd-p sinp2£ c o s ^ = B.

The values1 of ^ and h are derived immediately from the knownresult

_ r + a

J - 0dxe~x\

The last equation is in fact an identity, and consequently doesnot cease to be so, when we substitute for a; the quantity

y

The substitution gives

v ^ = 7^ / fy e~y" "1 = -.•=— Idy (COsy2 — V — 1 sin y2).

Thus the real part of the second member of the last equationis JIT and the imaginary part nothing. Whence we conclude

N/TT = ~7| (jdy cos y2 + jdy sin y2) >

1 More readily from the known results given in § 360, viz.—

du sin u /^r duT=~~ — \f 9 - ^ e t u ~z2> 2 -r= - ^ ^ then

dz sin «?=J yy/^. and | + " tfe sin 32=2 T (?.: sin ««= yy/^.

B.. for the cosine from r * L ° ^ ? « A r p T _, .

SECT. IV.J VALUES OF THE INTEGRALS. 415

and 0 = J dy cos y2 - j dy sin y \

or J dycosy2 = ff = A>J~, Jdy$my2 = h = A^?.

It remains only to determine, by means of the equations (a)and (6), the values of the two integrals

/ dy cos y2 cos 2by and | dy sin y2 sin 2by.

They can be expressed thus:

A = j dy cos y2 cos 2by = h sin b2 4- g cos b2,

B = I dy sin y2 cos 2fo/ = h cos 52— ^ sin b2;

whence we conclude

JK 1 / z2

writing sin - , or cos instead of \J -, we have

= ^ s i n ^ - - | ^ (2).and

408. The proposition expressed by equation (B) Article 404,or by equation (E) Article 361, which has served to discover theintegral (S) and the preceding integrals, is evidently applicable toa very great number of variables. In fact, in the general equation

= 2^J ^zfi*)] ^dp cos (px-pa),M

or f (®) = ?- <%> I <** cos

416 THEOBY OF HEAT. [CHAP. IX.

we can regard f(x) as a function of the two variables x and y.The function/(a) will then be a function of a and y. We shallnow regard this function / (a, y) as a function of the variable y,and we then conclude from the same theorem (B)f Article 404,

that /(a, y) = jj (a, /3) fdq cos (qy - qfi).

We have therefore, for the purpose of expressing any functionwhatever of the two variables x and y, the following equation

c o s ^ ~

f dq cos (qy-q/3)...(BB).J - G O

We form in the same manner the equation which belongs tofunctions of three variables, namely,

jdp cos (px —poi) jdq cos (jy — q/3) jdr cos (?^ — ry) (BBS),

each of the integrals being taken between the limits — 00and + co.

It is evident that the same proposition extends to functionswhich include any number whatever of variables. It remains toshow how this proportion is applicable to the discovery of theintegrals of equations which contain more than two variables.

409. For example, the differential equation being

d?v d2v d?v

we wish to ascertain the value of v as a function of (x, y, t), suchthat; 1st, on supposing t = 0, v or / (# , y} f) becomes an arbitraryfunction </> (x, y) of x and y\ 2nd, on making t = 0 in the value

of -J-, or / ' (x,y, t), we find a second entirely arbitrary function

SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 417

From the form of the differential equation (c) we can inferthat the value of v which satisfies this equation and the two pre-ceding conditions is necessarily the general integral. To discoverthis integral, we first give to v the particular value

i) = cos mt cos px cos qy.

The substitution of v gives the condition m = Jp2 + q2

It is no less evident that we may write

v = cosp (x — a) cos q(y — ft) cos t Jp2 + q*,

or

v= Ida. I dft F (a, ft) jdp cos (px - pa) /dj cos (22/ - qfi) cos f Vp" + 2*>

whatever be the quantities p} q} a, ft and F (a, /3), which containneither x, y, nor £. In fact this value of t is merely the sum ofparticular values.

If we suppose t = 0, v necessarily becomes <j> (x, y). We havetherefore

$ {x> y) = \doL ld/3 F(a, ft) jdp cos (px —pz) \dq cos (qy — qft).

Thus the problem is reduced to determining F(a9ft), so thatthe result of the indicated integrations may be cj> (%, y). Now, oncomparing the last equation with equation (BB), we find

0 fo 2/) = {^) ]_d*J dft <j> (a, £) J ^ cos (pa? -pa )

/ dq cos (qy-qft).J - c o

Hence the integral may be expressed thus:

> ft)jdpcos(px-p7.)jdqcos(qy-qft)costjp*+q2.

We thus obtain a first part u of the integral; and, denotingby W the second part, which ought to contain the other arbitraryfunction i/r (x} y), we have

V = U + W,F. H. 27

and we must take W to be the integral judt, changing only (x, y), when t is made= 0; and at the same time W becomes nothing, since the integra-tion, with respect to t, changes the cosine into a sine.

Further, if we take the value of -y-, and make £ = 0, the firstetc

part, which then contains a sine, becomes nothing, and thesecond part becomes equal to yjr (x, y). Thus the equationv = u+ IF is the complete integral of the proposed equation.

We could form in the same manner the integral of theequation

cPv <Pv d?v d2v

It would be sufficient to introduce a new factor

1 / x^_ cos (rz - ry) ,

and to integrate with respect to r and 7.

ATK T . ,1 J ±- i drv d2v d:v ~ ., .410. Let the proposed equation be -j—2 4- -j—2 + - — = 0 ; it is

doc ^y d'Z

required to express v as a function f(x,y,z), such that, 1st,f(x,y,0) may be an arbitrary function <p(x,y); 2nd, that onmaking z = 0 in the function -7- f(x,y,z) we may find a second

arbitrary function yfr (#, y). It evidently follows, from the form ofthe differential equation, that the function thus determined willbe the complete integral of the proposed equation.

To discover this equation we may remark first that the equa-tion is satisfied by writing v = cos^# cos qy emz, the exponentsp and q being any numbers whatever, and the value of m being

± JtfTtf.We might then also write

v = cos (px - pa) cos (qy - qfi) (e^^7^2 -f e~

SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 419

or

v = jd2Jdl3F(oii fi)jdpjdq cos (px -px) cos (qy - q/3)

If z be made equal to 0, we have, to determine F(oty /3), thefollowing condition

0 (Xf y) = jdx I dfi F (a, ft) I dp jdq cos (px -pa) cos (qy - q/3);

and, on comparing with the equation (BB), we see that

we have then, as the expression of the first part of the integral,

u = (^j jdzjdfi <f> (a, /3) jdp cos (px -px) Jdq cos (qy - qB)

The value of u reduces to <f> (x, y) when z = 0, and the same

substitution makes the value of ~j- nothing.

We might also integrate the value of u with respect to z, andgive to the integral the following form in which | is a newarbitrary function:

( 1 \2 f f f fJr=f^-J Ida ld/3yjr (a,/3) jdp cos (px — pj) dqcos (qy- q{3)

The value of W becomes nothing >when z ~ 0, and the same

substitution makes the function —7— equal to i/r (x, y). Hence

the general integral of the proposed equation is v = v. + W.

411. Lastly, let the equation be

cPv d*v _ dlv d'v A

it is required to determine v a s a function/(a?, y, t), which satisfiesthe proposed equation (e) and the two following conditions:namely, 1st, the substitution t = 0 in f(x, y, t) must give anarbitrary function <f> (at, y); 2nd, the same substitution in

-j f(x, y, t) must give a second arbitrary function ^ (x, y).

I t evidently follows from the form of equation (e), and fromthe principles which we have explained above, that the function v,when determined so as to satisfy the preceding conditions, will bethe complete integral of the proposed equation. To discover thisfunction we write first,

v = cos px cos qy cos mt,

whence we derive

cPv o d4v 4 d*v „ o d4v A

We have then the condition m=p2 + q2. Thus we can write

v = cospx cos qy cos t (p2 + q2),

or v = cos (px —pot.) cos (qy — q$) cos (pH + qH),

or v=\d% Id/3F(a, /3) jdp \dq cos {px — pa) cos (qy — q/3)

cos (pH + ^ ) -

When we make t = 0, we must have v = <> (x, y); which servesto determine the function F(a, jS). If we compare this with thegeneral equation (BE), we find that, when the integrals are taken

/ 1 \ 2

between infinite limits, the value of F(a, /3) is (— j <f> (a, /3). We

have therefore, as the expression of the first part u of theintegral,

= 12 \d\d^^ ( ^) J d j d ( ~ P * ) cos (?y -cos (pH + $2^).

Integrating the value of M with respect to t, the second arbi-trary function being denoted by yfr, we shall find the other partI F of the integral to be expressed thus :

SECT. IV.J OTHER FORM OF INTEGRAL. 4 2 1

W = Q^jfafa f (a> ft JdP]d2 cos (Px ~P^ cos I® ~ 2ftsin (p*t + qH)

? + <f 'If we make t = 0 in u and in W, the first function becomes

equal to </> (#,#), and the second nothing; and if we also make

£ = 0 in -j-u and in y- W, the first function becomes nothing,dt dt

and the second becomes equal to yjr [x,y): hence v = u +W is thegeneral integral of the proposed equation.

412. We may give to the value of u a simpler form by effect-ing the two integrations with respect to p and q. For thispurpose we use the two equations (1) and (2) which we haveproved in Art. 407, and we obtain the following integral,

Denoting by u the first part of the integral, and by IF thesecond, which ought to contain another arbitrary function, wehave

ctdt u and v = u •

i oJo

If we denote by /-t and v two new unknowns, such that wehave

oc — x _ /3 — y __

i 7 ^ YrV

and if we substitute for a, /3, dz, d/3 their values

we have this other form of the integral,

v = - dpi dv sin (/A2 + v*) <j> (x + 2/i Ji, y + 2^ V - 0 0 ^-QO

We could not multiply further these applications of ourformulae without diverging from our chief subject. The precedingexamples relate to physical phenomena, whose laws were un-known and difficult to discover; and we have chosen them because

the integrals of these equations, which have hitherto beenfruitlessly sought for, have a remarkable analogy with those whichexpress the movement of heat.

413. We might also, in the investigation of the integrals,consider first series developed according to powers of one variable,and sum these series by means of the theorems expressed by theequations (B), {BE). The following example of this analysis,taken from the theory of heat itself, appeared to us to beworthy of notice.

We have seen, Art. 399, that the general value of u derivedfrom the equation

dv d2v N( )

developed in series, according to increasing powers of the variablet, contains one arbitrary function only of x; and that when de-veloped, in series according to increasing powers of x, it containstwo completely arbitrary functions of t.

The first series is expressed thus:

The integral denoted by (/3), Art. 397; or

v = — \dz$ (a) \dp e~p2t cos (px —pd),

represents the sum of this series, and contains the single arbitraryfunction <f> (#).

The value of v, developed according to powers of xy containstwo arbitrary functions f(t) and F(t), and is thus expressed :

(X).

There is therefore, independently of equation (/S), anotherform of the integral which represents the sum of the last series,and which contains two arbitrary functions, f(t) and F(t).

SECT. IV.] SECONDARY INTEGRAL OF LINEAR EQUATION. 4 2 3

It is required to discover this second integral of the proposedequation, which cannot be more general than the preceding,but which contains two arbitrary functions.

We can arrive at it by summing each of the two series whichenter into equation (X). Now it is evident that if we knew, inthe form of a function of x and t, the sum of the first series whichcontains/^), it would be necessary, after having multiplied it bydx, to take the integral with respect to x, and to change f(t) intoF (t). We should thus find the second series. Further, it wouldbe enough to ascertain the sum of the odd terms which enter intothe first series: for, denoting this sum by JUL, and the sum of allthe other terms by v, we have evidently

rx rxv — \ dx I d

Jo Jo

It remains then to find the value of //,. Now the functionf(t) may be thus expressed, by means of the general equation (B),

f (?)= Tfr faf(«)jdp cos (pt-p*) (B).

It is easy to deduce from this the values of the functions

It is evident that differentiation is equivalent to writing in

the second member of equation (B), under the sign I dp, the

respective factors —]?, +p4, —pQ, &c

We have then, on writing once the common factor cos (pt—pz)y

Thus the problem consists in finding the sum of the serieswhich enters into the second member, which presents no difficulty.In fact, if y be the value of this series, we conclude

&y 2 >P*X* fix* , o d*y a

ch^-r+Y ~W+ y ords = -^'


Integrating this linear equation, and determining the arbitraryconstants, so that, when x is nothing, y may be 1, and

dy fry d?yTx> doc2' d?y

may be nothing, we find, as the sum of the series,

y = \ c o s x

It would be useless to refer to the details of this investigation;it is sufficient to state the result, which gives, as the integralsought,

v = - Uxf(z) jdq q jcos 2q* (* - a) (e** + e-*x) cos qx

- sin 2q2 (t- a) (e«*- e~^) sin qx\ + W. (/3/3).

The term W is the second part of the integral; it is formed byintegrating the first part with respect to x, from x = 0 to x = x,and by changing f into F. Under this form the integral containstwo completely arbitrary functions f(t) and F (t). If, in the valueof v, we suppose x nothing, the term W becomes nothing byhypothesis, and the first part u of the integral becomes f(t). If

(ID

we make the same substitution x = 0 in the value of ^- it isax

evident that the first part -j- will become nothing, and that thedW

second, -j—, which differs only from the first by the function

F being substituted for / , will be reduced to F (t). Thus theintegral expressed by equation (/3/3) satisfies all the conditions,and represents the sum of the two series which form the secondmember of the equation (X).

This is the form of the integral which it is necessary to selectin several problems of the theory of heat1; we see that it is verydifferent from that which is expressed by equation (J3)9 Art. 397.

1 See the article by Sir W. Thomson, " On the Linear Motion of Heat," Part II.Art. 1. Camb. Math. Journal, Vol. III. pp. 206—8. [A.F.]

SECT. IV.] SECONDARY INTEGRAL OF LINEAR EQUATION. 425

414. We may employ very different processes of investigationto express, by definite integrals, the sums of series which repre-sent the integrals of differential equations. The form of theseexpressions depends also on the limits of the definite integrals.We will cite a single example of this investigation, recalling theresult of Art. 311. If in the equation which terminates thatArticle we write % + tsinit under the sign of the function <£,we have

^ f

Denoting by v the sum of the series which forms the secondmember, we see that, to make one of the factors 22, 42, 62, &c.disappear in each term, we must differentiate once with respectto t, multiply the result by t} and differentiate a second time withrespect to t. We conclude from this that v satisfies the partialdifferential equation

d2v d2v 1 dv

We have therefore, to express the integral of this equation,

1 fn

v = ~ I du <£ (x +1 sin u) -f W.

The second part W of the integral contains a new arbitraryfunction.

The form of this second part W of the integral differs verymuch from that of the first, and may also be expressed by definiteintegrals. The results, which are obtained by means of definiteintegrals, vary according to the processes of investigation by whichthey are derived, and according to the limits of the integrals.

415. It is necessary to examine carefully the nature of thegeneral propositions which serve to transform arbitrary functions;for the use of these theorems is very extensive, and we derivefrom them directly the solution of several important physicalproblems, which could be treated by no other method. The


following proofs, which we gave in our first researches, are verysuitable to exhibit the truth of these propositions.

In the general equation

I r+* r+QO

f (x) = - I deaf (a) I dp cos (pz. -poo),

which is the same as equation (B), Art. 404, we may effect the in-tegration with respect to p, and we find

a-xWe ought then to give to p, in the last expression, an infinite

value; and, this being done, the second member will express thevalue of f(x). We shall perceive the truth of this result bymeans of the following construction. Examine first the definite

integral I dx , which we know to be equal to \*7r, Art. 356.Jo ®

If we construct above the axis of x the curve whose ordinate is.

sin x, and that whose ordinate is - , and then multiply the ordinatex

of the first curve by the corresponding ordmate of the second, wemay consider the product to be the ordmate of a third curvewhose form it is very easy to ascertain.

Its first ordinate at the origin is 1, and the succeeding ordinatesbecome alternately positive or negative; the curve cuts the axisat the points where x = w, 2TT, 3TT, &C., and it approaches nearerand nearer to this axis.

A second branch of the curve, exactly like the first, is situatedr00 sin x

to the left of the axis of y. The integral / dx is the areaJo x

included between the curve and the axis of xy and reckoned fromx = 0 up to a positive infinite value of x.

The definite integral / dx —±-—, in which p is supposed to bea Jo X r rx-

any positive number, has the same value as the preceding. In-00

fact, let px = z\ the proposed integral will become I dz ^ ^ - , and,Jo z

consequently, it is also equal to \ir. This proposition is true,

SECT. IV.] AREAS REPRESENTING INTEGRALS. 427

whatever positive number p may be. If we suppose, for example,, .. sin lOx T . ...

p = 10, the curve whose ordmate is has sinuosities very* x

much closer and shorter than the sinuosities whose ordinate is

??_?• but the whole area from x = 0 up to x = oo is the same.

Suppose now that the number p becomes greater and greater,and that it increases without limit, that is to say, becomes infinite.

The sinuosities of the curve whose ordinate is -—— are infinitely

TT

near. Their base is an infinitely small length equal to — . That

being so, if we compare the positive area which rests on one

of these intervals — with the negative area which rests on thefollowing interval, and if we denote by Xthe finite and sufficientlylarge abscissa which answers to the beginning of the first arc,we see that the abscissa x, which enters as a denominator into

s i n T)octhe expression ^ of the ordmate, has no sensible variation in

xthe double interval —, which serves as the base of the two areas.

p

Consequently the integral is the same as if x were a constantquantity. It follows that the sum of the two areas which succeedeach other is nothing.

The same is not the case when the value of x is infinitely2TT

Psmall, since the interval — has in this case a finite ratio to the

value of x. We know from this that the integral / dx- — , inJo x

which we suppose jp to be an infinite number, is wholly formed outof the sum of its first terms which correspond to extremely smallvalues of x. When the abscissa has a finite value X, the areadoes not vary, since the parts which compose it destroy each othertwo by two alternately. We express this result by writing

I"*Jo


The quantity a>, which denotes the limit of the second integral,has an infinitely small value; and the value of the integral is thesame when the limit is ay and when it is oo.

416. This assumed, take the equation

*, * 1 [+°° , , / N s i n p ( a — x) . N

/ ( # ) = - dxf(a)—^ -, (/? = oo).

Having laid down the axis of the abscissae a, construct abovethat axis the curve ff, whose ordinate is / ( a ) . The form ofthis curve is entirely arbitrary; it might have ordinates existingonly in one or several parts of its course, all the other ordinatesbeing nothing.

Construct also above the same axis of abscissae a curved line ss

whose ordinate is — — , z denoting the abscissa and p a veryz

great positive number. The centre of this curve, or the pointwhich corresponds to the greatest ordinate p, may be placed at theorigin 0 of the abscissae a, or at the end of any abscissa whatever.We suppose this centre to be successively displaced, and to betransferred to all points of the axis of a, towards the right, depart-ing from the point 0. Consider what occurs in a certain positionof the second curve, when the centre has arrived at the point x,which terminates an abscissa x of the first curve.

The value of x being regarded as constant, and a being theonly variable, the ordinate of the second curve becomes

sin p(rJL — x)a — x

If then we link together the two curves, for the purpose offorming a third, that is to say, if we multiply each ordinate of thesecond, and represent the product by an ordinate of a third curvedrawn above the axis of a, this product is

a — xThe whole area of the third curve, or the area included between

this curve and the axis of abscissae, may then be expressed by

sin p (a — x)a — xc

SECT. IV.] EXAMINATION OF AN INTEGRAL. 429

Now the number p being infinitely great, the second curve hasall its sinuosities infinitely near; we easily see that for all pointswhich are at a finite distance from the point x, the definiteintegral, or the whole area of the third curve, is formed of equalparts alternately positive or negative, which destroy each other twoby two. In fact, for one of these points situated at a certain dis-tance from the point %, the value of/(a) varies infinitely little

when we increase the distance by a quantity less than — . Thesame is the case with the denominator a — x, which measures thatdistance. The area which corresponds to the interval — is there-fore the same as if the quantities/(a) and a — x were not variables.Consequently it is nothing when a — x is a finite magnitude.Hence the definite integral may be taken between limits as nearas we please, and it gives, between those limits, the same result asbetween infinite limits. The whole problem is reduced then totaking the integral between points infinitely near, one to the left,the other to the right of that where a — x is nothing, that is to sayfrom <x = x — w to a = x-\-co} denoting by co a quantity infinitelysmall. In this interval the function f (a) does not vary, it isequal to/(a?), and may be placed outside the symbol of integra-tion. Hence the value of the expression is the product of f(x) by

J a — x

taken between the limits a — x = — o>, and a — x = to.

Now this integral is equal to 7r, as we have seen in the pre-ceding article ; hence the definite integral is equal to irf{x)i whencewe obtain the equation

- . . 1 p"00 . . 2 sin p (« — x)

1 / -+0 0 /•+*>KZ dxf(a)\ dpcos(px-px) (B).

417. The preceding proof supposes that notion of infinitequantities which has always been admitted by geometers. Itwould be easy to offer the same proof under another form, examin-ing the changes which result from the continual increase of the

factor p under the symbol sin p (a — as). These considerations aretoo well known to make it necessary to recall them.

Above all, it must be remarked that the function f (x), to whichthis proof applies, is entirely arbitrary, and not subject to a con-tinuous law. We might therefore imagine that the enquiry isconcerning a function such that the ordinate which represents ithas no existing value except when the abscissa is included betweentwo given limits a and b, all the other ordinates being supposednothing; so that the curve has no form or trace except above theinterval from x = a to x = 6, and coincides with the axis of a inall other parts of its course.

The same proof shews that we are not considering here infinitevalues of x, but definite actual values. We might also examine onthe same principles the cases in which the function f(x) becomesinfinite, for singular values of x included between the given limits;but these have no relation to the chief object which we have inview, which is to introduce into the integrals arbitrary functions;it is impossible that any problem in nature should lead to thesupposition that the function f(x) becomes infinite, when wegive to x a singular value included between given limits.

In general the function/(.#) represents a succession of valuesor ordinates each of which is arbitrary. An infinity of values beinggiven to the abscissa x, there are an equal number of ordinates/ ' (x). All have actual numerical values, either positive or negativeor nul.

We do not suppose these ordinates to be subject to a commonlaw; they succeed each other in any manner whatever, and each ofthem is given as if it were a single quantity.

It may follow from the very nature of the problem, and fromthe analysis which is applicable to it, that the passage from oneordinate to the following is effected in a continuous manner. Butspecial conditions are then concerned, and the general equation (B),considered by itself, is independent of these conditions. It isrigorously applicable to discontinuous functions.

Suppose now that the function f(x) coincides with a certainanalytical expression, such as sin#, e~x\ or </> (x), when we give tox a value included between the two limits a and 6, and that all

SECT. IV.] FUNCTIONS COINCIDING BETWEEN LIMITS. 431

the values of f{x) are nothing when x is not included between aand b; the limits of integration with respect to a, in the precedingequation (B)y become then a = a, a = 6; since the result is the sameas for the limits a = — oo , a = GO , every value of cf> (a) being nothingby hypothesis, when a is not included between a and b. We havethen the equation

W = 2^j ^$(^1The second member of this equation (Bf) is a function of the

variable x\ for the two integrations make the variables a and p dis-appear, and x only remains with the constants a and b. Now thefunction equivalent to the second member is such, that on substitut-ing for x any value included between a and 6, we find the sameresult as on substituting this value of x in <p (x); and we find a nulresult if, in the second member, we substitute for x any value notincluded between a and h. If then, keeping all the other quantitieswhich form the second member, we replaced the limits a and bby nearer limits a and b\ each of which is included between a andb} we should change the function of x which is equal to the secondmember, and the effect of the change would be such that thesecond member would become nothing whenever we gave to x avalue not included between a and b''; and, if the value of x wereincluded between a and b\ we should have the same result ason substituting this value of x in <f>(x).

We can therefore vary at will the limits of the integral in thesecond member of equation (B'). This equation exists always forvalues of x included between any limits a and b, which we mayhave chosen; and, if we assign any other value to x, the secondmember becomes nothing. Let us represent <fi (x) by the variableordinate of a curve of which x is the abscissa ; the second member,whose value is/(#), will represent the variable ordinate of a secondcurve whose form will depend on the limits a and b. If theseHmits are — oo and + oo , the two curves, one of which has (f)(x) forordinate, and the other f(x), coincide exactly through the wholeextent of their course. But, jf we give other values a and b to theselimits, the two curves coincide exactly through every part of theircourse which corresponds to the interval from x = a to x = 6. Toright and left of this interval, the second curve coincides precisely


at every point with the axis of x. This result is very remarkable,and determines the true sense of the proposition expressed byequation (5).

418. The theorem expressed by equation (II) Art. 234 mustbe considered under the same point of view. This equationserves to develope an arbitrary function / (x) in a series of sines orcosines of multiple arcs. The function fix) denotes a functioncompletely arbitrary, that is to say a succession of given values,subject or not to a common law, and answering to all the values ofx included between 0 and any magnitude X.

The value of this function is expressed by the followingequation,

2f (a) coa^(x-a) (A).

The integral, with respect to a, must be taken between thelimits a = a, and a = b; each of these limits a and b is any quantitywhatever included between 0 and X. The sign 2 affects theinteger number i, and indicates that we must give to i everyinteger value negative or positive, namely,

. . . - 5 , - 4 , - 3 , - 2 , - 1 , 0, + 1 , +2 , + 3 , + 4 , + 5 , . . .

and must take the sum of the terms arranged under the sign 2.After these integrations the second member becomes a function ofthe variable x only, and of the constants a and b. The generalproposition consists in this : 1st, that the value of the secondmember, which would be found on substituting for x a quantityincluded between a and 6, is equal to that which would be obtainedon substituting the same quantity for x in the function/(a:); 2nd,every other value of x included between 0 and Xy but not includedbetween a and b> being substituted in the second member, gives amil result.

Thus there is no function / (# ) , or part of a function, whichcannot be expressed by a trigonometric series.

The value of the second member is periodic, and the intervalof the period is X, that is to say, the value of the second memberdoes not change when x + X is written instead of x. All itsvalues in succession are renewed at intervals X.

SECT. IV.] TRANSFORMATION OF FUNCTIONS. 433

The trigonometrical series equal to the second member isconvergent; the meaning of this statement is, that if we give tothe variable x any value whatever, the sum of the terms of theseries approaches more and more, and infinitely near to, a definitelimit. This limit is 0, if we have substituted for x a quantityincluded between 0 and X, but not included between a and b;but if the quantity substituted for x is included between a and b,the limit of the series has the same value as fix). The lastfunction is subject to no condition, and the line whose ordinate itrepresents may have any form; for example, that of a contourformed of a series of straight lines and curved lines. We see bythis that the limits a and b, the whole interval X> and the natureof the function being arbitrary, the proposition has a very exten-sive signification; and, as it not only expresses an analyticalproperty, but leads also to the solution of several importantproblems in nature, it was necessary to consider it under differentpoints of view, and to indicate its chief applications. We havegiven several proofs of this theorem in the course of this work.That which we shall refer to in one of the following Articles(Art. 424) has the advantage of being applicable also to non-periodic functions.

If we suppose the interval X infinite, the terms of the seriesbecome differential quantities; the sum indicated by the sign 2becomes a definite integral, as was seen in Arts. 353 and 355, andequation (A) is transformed into equation (B). Thus the latterequation (B) is contained in the former, and belongs to the casein which the interval X is infinite: the limits a and b are thenevidently entirely arbitrary constants.

419. The theorem expressed by equation (B) presents alsodivers analytical applications, which we could not unfold withoutquitting the object of this work; but we will enunciate theprinciple from which these applications are derived.

We see that, in the second member of the equation

the function f(x) is so transformed, that the symbol of thefunction / affects no longer the variable x, but an auxiliary

F. H. 28


variable a. The variable x is only affected by the symbol cosine.It follows from this, that in order to differentiate the function/(.r)with respect to x, as many times as we wish, it is sufficient todifferentiate the second member with respect to x under thesymbol cosine. We then have, denoting by % any integer numberwhatever,

We take the upper sign when i is even, and the lower signwhen i is odd. Following the same rule relative to the choiceof sign

d2l+1 i r r(a) \dpp2l+ sin (px — j

We can also integrate the second member of equation (5)several times in succession, with respect to x; it is sufficient towrite in front of the symbol sine or cosine a negative powerof p.

The same remark applies to finite differences and to summa-tions denoted by the sign 2, and in general to analytical operationswhich may be effected upon trigonometrical quantities. The chiefcharacteristic of the theorem in question, is to transfer the generalsign of the function to an auxiliary variable, and to place thevariable x under the trigonometrical sign. The function f(x)acquires in a manner, by this transformation, all the properties oftrigonometrical quantities ; differentiations, integrations, and sum-mations of series thus apply to functions in general in the samemanner as to exponential trigonometrical functions. For whichreason the use of this proposition gives directly the integralsof partial differential equations with constant coefficients. Infact, it is evident that we could satisfy these equations by par-ticular exponential values; and since the theorems of which weare speaking give to the general and arbitrary functions thecharacter of exponential quantities, they lead easily to the expres-sion of the complete integrals.

The same transformation gives also, as we have seen inArt. 413, an easy means of summing infinite series, when theseseries contain successive differentials, or successive integrals of the

SECT. IV.] REAL AND UNREAL PARTS OF A FUNCTION. 435

same function; for the summation of the series is reduced, bywhat precedes, to that of a series of algebraic terms.

420. We may also employ the theorem in question for thepurpose of substituting under the general form of the function abinomial formed of a real part and an imaginary part. Thisanalytical problem occurs at the beginning of the calculus ofpartial differential equations; and we point it out here since ithas a direct relation to our chief object.

If in the function f(x) we write fi + v J — 1 instead of x} theresult consists of two parts <f> + J — 1 ty. The problem is todetermine each of these functions (f> and ty in terms of p and v.We shall readily arrive at the result if we replace fQr) by theexpression

df ()Jd cos (pat-py),

for the problem is then reduced to the substitution of /JL + V J — 1instead of x under the symbol cosine, and to the calculation of thereal term and the coefficient of J — 1. We thus have

/(*) =/G* + V J^l) = ~jch (0L)Jdp COS [p {fJL - ft) + pv J~-l]

= 4^jd*f(*) fdP (cos (Pf* -P*) (ePV + e~pv)W ^ s i n (pfi-pa) (ePv-e-Pv)};

hence 0 = — jdotf(%) I dp cos (pp-pz) (epv + e~^%

yfr = — ldxf(a) I dp sin (pp -pi) (e*v - e"^v).

Thus all the functions fix) which can be imagined, even thosewhich are not subject to any law of continuity, are reduced to theform M+Nj- 1, when we replace the variable x in them by thebinomial fi+v J^1.

28 2

421. To give an example of the use of the last two formulae,cPv d2v

let us consider the equation -j-2 + - ^ = 0, which relates to the

uniform movement of heat in a rectangular plate. The generalintegral of this equation evidently contains two arbitrary func-tions. Suppose then that we know in terms of x the value of vwhen y = 0, and that we also know, as another function of x, the

dvvalue of -T when y = 0, we can deduce the required integral from

aythat of the equation

which has long been known; but we find imaginary quantitiesunder the functional signs: the integral is

The second part W of the integral is derived from the first byintegrating with respect to yy and changing (f> into ty.

It remains then to transform the quantities <$>{% +y J— 1) and$(x — yj—l), in order to separate the real parts from the ima-ginary parts. Following the process of the preceding Article wefind for the first part u of the integral,

1 /•+<*> / • + »

u = — I d(xf(a) I dp cos (px —po) (e

and consequently

C j+X ^ cos (px-pz) (e**-

e~py)y

The complete integral of the proposed equation expressed inreal terms is therefore v = u 4- W; and we perceive in fact,1st, that it satisfies the differential equation ; 2nd, that on makingy = 0 in it, it gives v=f(x); 3rd, that on making y = 0 in the

function -j- . the result is Fix).dy v J

SECT. IV.] DIFFERENTIATION OF FUNCTIONS. 437

422. We may also remark that we can deduce from equation(2?) a very simple expression of the differential coefficient of the

ith order, - T - I / ( # ) , or of the integral I dxlf{x).

The expression required is a certain function of x and of theindex i. It is required to ascertain this function under a formsuch that the number i may not enter it as an index, but as aquantity, in order to include, in the same formula, every case inwhich we assign to i any positive or negative value. To obtain itwe shall remark that the expression

cos

iir . . iiror cos r cos =— sin r sin -~-,

becomes successively

- sin r, — cos r, + sin r, + cos r, . — sin r, &c,

if the respective values of i are 1, 2, 3, 4, 5, &c. The same resultsrecur in the same order, when we increase the value of i. In thesecond member of the equation

f(p) = 2^jd*f^) JdP cos (jpx -pi),

we must now write the factor p* before the symbol cosine, and

add under this symbol the term + %•= . We shall thus have

dl 1 f+3

da? = Yir J7r

C0S \PX ~ + i

The number i9 which enters into the second member, may beany positive or negative integer. We shall not press these applica-tions to general analysis; it is sufficient to have shewn the use ofour theorems by different examples. The equations of the fourthorder, (d)f Art. 405, and (V), Art. 411, belong as we have said todynamical problems. The integrals of these equations were notyet known when we gave them in a Memoir on the Vibrations of


Elastic Surfaces, read at a sitting of the Academy of Sciences1,6th June, 1816 (Art. VI. §§ 10 and 11, and Art. VII. §§ 13 and 14).They consist in the two formulae S and S', Art. 406, and in the twointegrals expressed, one by the first equation of Art. 412, the otherby the last equation of the same Article. We then gave severalother proofs of the same results. This memoir contained also theintegral of equation (c), Art. 409, under the form referred to inthat Article. With regard to the integral (/3/3) of equation (a),Art. 413; it is here published for the first time.

423. The propositions expressed by equations (A) and (B')7

Arts. 418 and 417, may be considered under a more general pointof view. The construction indicated in Arts. 415 and 416 applies

not only to the trigonometrical function — — ^ - ; but suitscc — cc

all other functions, and supposes only that when the number pbecomes infinite, we find the value of the integral with respect toa, by taking this integral between extremely near limits. Nowthis condition belongs not only to trigonometrical functions, but isapplicable to an infinity of other functions. We thus arrive atthe expression of an arbitrary function f(x) under different veryremarkable forms; but we make no use of these transformationsin the special investigations which occupy us.

With respect to the proposition expressed by equation (A),Art. 418, it is equally easy- to make its truth evident by con-structions, and this was the theorem for which we employed themat first. It will be sufficient to indicate the course of the proof.

1 The date is inaccurate. The memoir was read on June 8th, 1818, as appearsfrom an abstract of it given in the Bulletin des Sciences par la Societe Philomatique,September 1818, pp. 129—136, entitled, Note relative aux vibrations des surfaceselastiqucs ct au mouvement des ondes, par M. Fourier. The reading of the memoirfurther appears from the Analyse des travaux de VAcademie des Sciences pendantVannee 1818, p. xiv, and its not having been published except in abstract, from aremark of Poisson at pp. 150—1 of his memoir Sur les equations aux differencespartlelles, printed in the Memoires de V Academie des Sciences, Tome in. (year 1818),Paris, 1820. The title, Memoire sur les vibrations des surfaces Slastiquest parM. Fourier, is given in the Analyse, p. xiv. The object, " to integrate severalpartial differential equations and to deduce from the integrals the knowledge of thephysical phenomena to which these equations refer," is stated in the Bulletin,P. 120. [A. F.]

SECT. IV.J EXAMINATION OF AN INTEGRAL. 439

In equation (A), namely,

c o s

we can replace the sum of the terms arranged under thesign 2 by its value, which is derived from known theorems.We have seen different examples of this calculation previously,Section III., Chap. in. It gives as the result if we suppose,in order to simplify the expression, 2-TT — X, and denote a-xby r,

S+j . . . . sin r

cos ir = cos ir + sin ir : — .-j J J J versm r

We must then multiply the second member of this equationby da/(a), suppose the number j infinite, and integrate froma = — IT to a = + ?r. The curved line, whose abscissa is a andordinate cos V, being conjoined with the line whose abscissa isa and ordinate / ( a ) , that is to say, when the correspondingordinates are multiplied together, it is evident that the area ofthe curve produced, taken between any limits, becomes nothingwhen the number j increases without limit. Thus the first termcos jr gives a nul result.

The same would be the case with the term sin jr, if it were

not multiplied by the factor : — ; but on comparing ther J versm r

three curves which have a common abscissa a, and as ordinatessin r

sin jr, : , /(a), we see clearly that the integralJ versm r J v J J &

Woo versm r

has no actual values except for certain intervals infinitely small,

namely, when the ordinate :— becomes infinite. This willJ versm r

take place if r or a — x is nothing; and in the interval in whichOL differs infinitely little from x, the value o f / (a ) coincides withf(x). Hence the integral becomes

2/0) f° dr sin jr^, or 4/(a) [ -^ sinjr,


which is equal to 2irf(x), Arts. 415 and 356. Whence we con-clude the previous equation [A).

When the variable x is exactly equal to — IT or +7r, the con-struction shews what is the value of the second member of theequation (A)y [ i / ( - 7 r ) or £ / (TT)] .

If the limits of integrations are not — IT and + IT, but othernumbers a and b, each of which is included between — IT and-f 7r, we see by the same figure what the values of x are, for whichthe second member of equation [A) is nothing.

If we imagine that between the limits of integration certainvalues of f(a) become infinite, the construction indicates in whatsense the general proposition must be understood. But we donot here consider cases of this kind, since they do not belongto physical problems.

If instead of restricting the limits — IT and + 7r, we givegreater extent to the integral, selecting more distant limits aand b'} we know from the same figure that the second memberof equation {A) is formed of several terms and makes the resultof integration finite, whatever the function f{x) may be.

fJL —— If*

We find similar results if we write 2TT —^~ instead of r, the

limits of integration being — X and + X.It must now be considered that the results at which we

have arrived would also hold for an infinity of different functionsof sin jr. It is sufficient for these functions to receive valuesalternately positive and negative, so that the area may becomenothing, when j increases without limit. We may also vary

SI Y\ 7*the factor •--.—-, as well as the limits of integration, and we

versm r ^

may suppose the interval to become infinite. Expressions ofthis kind are very general, and susceptible of very different forms.We cannot delay over these developments, but it was necessaryto exhibit the employment of geometrical constructions; forthey solve without any doubt questions which may arise on theextreme values, and on singular values; they would not haveserved to discover these theorems, but they prove them and guideall their applications.

SECT. IV.] DEVELOPMENT IN SERIES OF FUNCTIONS. 441

424. We have yet to regard the same propositions underanother aspect. If we compare with each other the solutionsrelative to the varied movement of heat in a ring, a sphere, arectangular prism, a cylinder, we see that we had to developean arbitrary function f(x) in a series of terms, such as

The function </>, which in the second member of equation(A) is a cosine or a sine, is replaced here by a function whichmay be very different from a sine. The numbers fiv fi2, /JL3, &C.instead of being integers, are given by a transcendental equation,all of whose roots infinite in number are real.

The problem consisted in finding the values of the coefScientsav a2y a3...ai; they have been arrived at by means of definiteintegrations which make all the unknowns disappear, except one.We proceed to examine specially the nature of this process, andthe exact consequences which flow from it.

In order to give to this examination a more definite object,we will take as example one of the most important problems,namely, that of the varied movement of heat in a solid sphere.We have seen, Art. 290, that, in order to satisfy the initial dis-tribution of the heat, we must determine the coefficients av aiy

az... ai5 in the equation

xF(x) = aL sin (jixx) + a2 sin {[x2x) + az sin (JJLBX) + &C (e).

The function F{x) is entirely arbitrary; it denotes the valuev of the given initial temperature of the spherical shell whoseradius is x. The numbers fiv /JL2 ... \xi are the roots yit, of thetranscendental equation

tan

X is the radius of the whole sphere; h is a known numerical co-efficient having any positive value. We have rigorously proved inour earlier researches, that all the values of /JL or the roots of theequation ( / ) are real1. This demonstration is derived from the

1 The Mtmoires de VAcademie des Sciences, Tome x, Paris 1831, pp. 119—146,eontain Remarques generates sttr Vapplication des principes de Vanalyse algebrique

4 4 2 THEORY OF HEAT. [CHAP. IX.

general theory of equations, and requires only that we shouldsuppose known the form of the imaginary roots which every equa-tion may have. We have not referred to it in this work, since itsplace is supplied by constructions which make the proposition moreevident. Moreover, we have treated a similar problem analytically,in determining the varied movement of heat in a cylindrical body(Art. 308). This arranged, the problem consists in discoveringnumerical values for av a2, a39...ai9 &c9 such that the secondmember of equation (e) necessarily becomes equal to xF(x)9 whenwe substitute in it for x any value included between 0 and thewhole length X.

To find the coefficient a., we have multiplied equation (e) bydx sin fi.or9 and then integrated between the limits x = 0, x = X,and we have proved (Art. 291) that the integral

'Xdx sin fife sin fijX

ohas a null value whenever the indices i and j are not the same;that is to say when the numbers /x. and fx. are two different rootsof the equation ( / ) . It follows from this, that the definite inte-gration making all the terms of the second member disappear,except that which contains ai9 we have to determine this coefficient,the equation

fX # rX

dx [x F (x) sin [x.x\ = a A dx sin /x^ sin p.®.' 0 Jo

Substituting this value of the coefficient a. in equation (e), wederive from it the identical equation (e),

fX

da. aF(a) sin/x.a' (0.

fJo

dft sin pJ3 sin /xJ3o

aux Equations transcendantes, by Fourier. The author shews that the imaginaryroots of sec x=0 do not satisfy the equation tan#=0, since for them, tan^= sj - 1.The equation tan x = 0 is satisfied only by the roots of sin x = 0, which are all real.It may be shewn also that the imaginary roots of sec x=0 do not satisfy the equationx-m tan# = 0, where m is less than 1, but this equation is satisfied only by theroots of the equation f(x) = x cos x - m sinx = 0, which are all real. For iffr+i(x), fr(x), fr-\(

x)i a r e three successive differential coefficients of f(x), the valuesof x which make / . (x) =0, make the signs of fr+1 (x) and /J._1 (x) different. Henceby Fourier's Theorem relative to the number of changes of sign of f(x) and itssuccessive derivatives, f(x) can have no imaginary roots. [A. F.J

SECT. IV.J WHAT TERMS MUST BE INCLUDED. 4 4 3

In the second member we must give to i all its values, that is tosay we must successively substitute for fin all the roots /UL of theequation (/) . The integral must be taken for a from a = 0 toa = X, which makes the unknown a disappear. The same is thecase with /3, which enters into the denominator in such a mannerthat the term sin^.a? is multiplied by a coefficient a. whose valuedepends only on Ar and on the index i. The symbol S denotesthat after having given to i its different values, we must writedown the sum of all the terms.

The integration then offers a very simple means of determiningthe coefficients directly; but we must examine attentively theorigin of this process, which gives rise to the following remarks.

1st. If in equation (e) we had omitted to write down part ofthe terms, for example, all those in which the index is an evennumber, we should still find, on multiplying the equation bydx sin //,.#, and integrating from x = 0 to x = X, the same value ofan which has been already determined, and we should thus forman equation which would not be true; for it would contain onlypart of the terms of the general equation, namely, those whoseindex is odd.

2nd. The complete equation (e) which we obtain, after havingdetermined the coefficients, and which does not differ from theequation referred to (Art. 291) in which we might make £=0 andv =/(#), is such that if we give to x any value included between0 and X, the two members are necessarily equal; but we cannotconclude, as we have remarked, that this' equality would hold, ifchoosing for the first member xF{%) a function subject to a con-tinuous law, such as sin x or cos x, wTe were to give to x a valuenot included between 0 and X. In general the resulting equation(e) ought to be applied to values of x, included between 0 and X.Now the process which determines the coefficient ai does notexplain why all the roots fit must enter into equation (e), norwhy this equation refers solely to values of x, included between 0and X.

To answer these questions clearly, it is sufficient to revert tothe principles which serve as the foundation of our analysis.

We divide the interval X into an infinite number n of parts

equal to dx, so that we have ndx = X, and writ ing/(#) instead ofxF{x),we denote hjf19f99fs...ft...fn9 the values of/(x), whichcorrespond to the values dx, 2dx, 3dx,... idx ... ndx, assigned tox;' we make up the general equation (e) out of a number n ofterms; so that n unknown coefficients enter into it, av a2, a3,...aj...an. This arranged, the equation (e) represents w equationsof the first degree, which we should form by substituting succes-sively for x, its n values dx, 2dx, 3dx,... ndx. This system of nequations contains fx in the first equation, f2 in the second, fs inthe third, fn in the wth. To determine the first coefficient av wemultiply the first equation by a1, the second by <r2, the third byer3, and so on, and add together the equations thus multiplied.The factors ax, a2, cr3,...<rn must be determined by the condition,that the sum of all the terms of the second members which containa2 must be nothing, and that the same shall be the case with thefollowing coefficients a3, av ...an. All the equations being thenadded, the coefficient ax enters only into the result, and we havean equation for determining this coefficient. We then multiplyall the equations anew by other factors p19 p2, p3,...pn respectively,and determine these factors so that on adding the n equations, allthe coefficients may be eliminated, except a2. We have then anequation to determine a2. Similar operations are continued, andchoosing always new factors, we successively determine all theunknown coefficients. Now it is evident that this process of elimi-nation is exactly that which results from integration between thelimits 0 and X. The series cr1, <r2, <T5J...crr of the first factors isdx sin (fju^dx), dx sin (/n^dx), dx sin (fifidx)... dx sin (^ndx). Ingeneral the series of factors which serves to eliminate all the co-efficients except ai9 is dx sin (ji$x), dx sin QJU. 2dx), dx sin (JJL. Sdx)...dx sin (fju.ndx); it is represented by the general term dx sin (/ a?),in which we give successively to x all the values

djCy 2dx} 3dx, ... ndx.

We see by this that the process which serves to determine thesecoefficients, differs in no respect from the ordinary process of elimi-nation in equations of the first degree. The number n of equationsis equal to that of the unknown quantities av a2) a3...an, and isthe same as the number of given quantities fvfvfZ"-fn* Thevalues found for the coefficients are those which must exist in

SECT. IV.] CONDITIONS OF DEVELOPMENT. 445

order that the n equations may hold good together, that is to sayin order that equation (e) may be true when we give to x one ofthese n values included between 0 and X; and since the numbern is infinite, it follows that the first member f (x) necessarily coin-cides with the second, when the value of x substituted in eachis included between 0 and X.

The foregoing proof applies not only to developments of theform

ax sin fax) + a2 sin (^x) + a3sin (ji3x) + ... + aisin fax,

it applies to all the functions </> (fax) which might be substitutedfor sin (jip), maintaining the chief condition, namely, that the

rxintegral I dx$ (fax) <£ (jijx) has a nul value, when i and j are

different numbers.

If it be proposed to develope/(#) under the form

*, N , a.cosx <3LCOS2,£ a.cosix o/ ( » = « + / • + 7

2 . o + . . . + T • +&c,bx sm x b2 sin 2x oi cos %x

the quantities fa, /x2, fjL3...fay &c. will be integers, and the con-dition

\ dx cos ( 2iri ^ j sin f 2irj -^1 = 0,

always holding when the indices i and j are different numbers, weobtain, by determining the coefficients av bv the general equation(II), page 206, which does not differ from equation (A) Art. 418.

425. If in the second member of equation (e) we omitted oneor more terms which correspond to one or more roots fa of theequation ( / ) , equation (e) would not in general be true. Toprove this, let us suppose a term containing p. and a. not to bewritten in the second member of equation (e), we might multiplythe n equations respectively by the factors

dx sin (jijdx), dx sin (jifidx), dx sin (/i.Sdx) ...dx sin (fi.ndx);

and adding them, the sum of all the terms of the second memberswould be nothing, so that not one of the unknown coefficientswould remain. The result, formed of the sum of the first members,


that is to say the sum of the values / , / 2 , / 3 . . . / a J multipliedrespectively by the factors

dx sin (fjLjdx), dx sin (fij2dx), dx sin (fxf3dx):.. dx sin (/A^idx),

would be reduced to zero. This relation would then necessarilyexist between the given q u a n t i t i e s / , / , / g . . . / ^ ; and they could notbe considered entirely arbitrary, contrary to hypothesis. If thesequantities / > / 2 > / } • • • / , have any values whatever, the relation inquestion cannot exist, and we cannot satisfy the proposed con-ditions by omitting one or more terms, such as a. sin (fijx) inequation (e).

Hence the function f(x) remaining undetermined, that is tosay, representing the system of an infinite number of arbitraryconstants which correspond to the values of x included between 0and X, it is necessary to introduce into the second member ofequation (e) all the terms such as a5 sin (fifx)9 which satisfy thecondition

xdx sin [ifc sin fxjx = 0,

o

the indices i and j being different; but if it happen that thefunction f(x) is such that the n magnitudes / , / > > / • • • / * areconnected by a relation expressed by the equation

xdx sin ^jxf(x) = 0,

o

it is evident that the term a, sin^/s might be omitted in the equa-tion (e).

Thus there are several classes of functions/(.?) whose develop-ment, represented by the second member of the equation (e), doesnot contain certain terms corresponding to some of the roots fi.There are for example cases in which we omit all the termswhose index is even; and we have seen different examples of thisin the course of this work. But this would not hold, if the func-t i o n / ^ ) had all the generality possible. In all these cases, weought to suppose the second member of equation (e) to be com-plete, and the investigation shews what terms ought to be omitted,since their coefficients become nothing.

SECT. IV.J SYSTEM OF QUANTITIES REPRESENTED. 447

426. We see clearly by this examination that the function f(x)represents, in our analysis, the system of a number n of separatequantities, corresponding to n values of x included between 0 andX, and that these n quantities have values actual, and consequentlynot infinite, chosen at will. All might be nothing, except one,whose value would be given.

It might happen that the series of the n values flff2,f3 ...fn

was expressed by a function subject to a continuous law, such asx or x3, sin x, or cos x, or in general $ (x); the curve line 0C0,whose ordinates represent the values corresponding to the abscissax, and which is situated above the interval from x = 0 to x = X,coincides then in this interval with the curve whose ordinate is</> (x), and the coefficients a19 a2> a3... an of equation (e) determinedby the preceding rule always satisfy the condition, that any valueof x included between 0 and X, gives the same result when substi-tuted in (j) (x), and in the second member of equation (e).

F(x) represents the initial temperature of the spherical shellwhose radius is x. We might suppose, for example, F(x) = bx,that is to say, that the initial heat increases proportionally to thedistance, from the centre, where it is nothing, to the surfacewhere it is bX. In this case xF(x) or / (x) is equal to bx2] andapplying to this function the rule which determines the coeffi-cients, bx2 would be developed in a series of terms, such as

a1 sin fax) + a2 sin fax) + a3 sin fax) + ... + an sin (finx).

Now each term sin (fax), when developed according to powersof x, contains only powers of odd order, and the function bx2 isa power of even order. It is very remarkable that this functionbx2, denoting a series of values given for the interval from 0to X, can be developed in a series of terms, such as ai sin fax).

We have already proved the rigorous exactness of theseresults, which had not yet been presented in analysis, and wehave shewn the true meaning of the propositions which expressthem. We have seen, for example, in Article 223, that thefunction cos# is developed in a series of sines of multiple arcs,so that in the equation which gives this development, the firstmember contains only even powers of the variable, and the secondcontains only odd powers. Reciprocally, the function sin x, into


which only odd powers enter, is resolved, Art. 225, into a seriesof cosines which contain only even powers.

In the actual problem relative to the sphere, the value ofxF{x) is developed by means of equation (e). We must then,as we see in Art. 290, write in each term the exponential factor,which contains t, and we have to express the temperature v9

which is a function of x and t9 the equation[XI dxsin (fai) OLF^J)

xv = 2 sin (ji^e-W*^ (E).\ d& sin fa® zinfaff)Jo

The general solution which gives this equation (E) is whollyindependent of the nature of the function F(x) since this functionrepresents here only an infinite multitude of arbitrary constants,which correspond to as many values of x included between 0and X.

If we supposed the primitive heat to be contained in a partonly of the solid sphere, for example, from x = 0 to x = \Xy

and that the initial temperatures of the upper layers were nothing,it would be sufficient to take the integral

between the limits x = 0 and x =

In general, the solution expressed by equation (E) suits allcases, and the form of the development does not vary according tothe nature of the function.

Suppose now that having written sin x instead of F(x) we havedetermined by integration the coefficients a., and that we haveformed the equation

x sin x = a1 sin fxxx + a2 sin /JL2X + az sin /z3# -f &c.

It is certain that on giving to x any value whatever includedbetween 0 and X, the second member of this equation becomesequal to x sin x; this is a necessary consequence of our process.But it nowise follows that on giving to a? a value not includedbetween 0 and X9 the same equality would exist. We see thecontrary very distinctly in the examples which we have cited, and,

SECT. IV.] SINGLE LAYER INITIALLY HEATED. 449

particular cases excepted, we may say that a function subject to acontinuous law, which forms the first member of equations of thiskind, does not coincide with the function expressed by the secondmember, except for values of x included between 0 and X.

Properly speaking, equation (e) is an identity, which existsfor all values which may be assigned to the variable x\ eachmember of this equation representing a certain analytical functionwhich coincides with a known function f{x) if we give to thevariable x values included between 0 and X. With respect to theexistence of functions, which coincide for all values of the variableincluded between certain limits and differ for other values, it isproved by all that precedes, and considerations of this kind are anecessary element of the theory of partial differential equations.

Moreover, it is evident that equations (e) and (E) apply notonly to the solid sphere whose radius is X, but represent, one theinitial state, the other the variable state of an infinitely extendedsolid, of which the spherical body forms part; and when in theseequations we give to the variable x values greater than X,they refer to the parts of the infinite solid which envelops thesphere.

This remark applies also to all dynamical problems which aresolved by means of partial differential equations.

427. To apply the solution given by equation (E) to the casein which a single spherical layer has been originally heated, allthe other layers having nul initial temperature, it is sufficient to

take the integral jdx sin (/z.a) oF (a) between two very near limits,

a. = r, and a = r + u, r being the radius of the inner surface of theheated layer, and u the thickness of this layer.

We can also consider separately the resulting effect of theinitial heating of another layer included between the limits r + uand r + 2u; and if we add the variable temperature due to thissecond cause, to the temperature which we found when the firstlayer alone was heated, the sum of the two1 temperatures is thatwhich would arise, if the two layers were heated at the same time.In order to take account of the two joint causes, it is sufficient to

F. H. 29

450 THEORY OF HEAT. [CHAP, IX.

take the integral p a sin (^a) aF(a) between the limits a = r and

a = r + 2ti. More generally, equation (JS) being capable of beingput under the form

[Xj r?t \ • ^ sin fiv — \ den, aF(a) smup 2 —75 —0 x I d/3 siin /JLJ3 sin ftJ3

we see that the whole effect of the heating of different layers isthe sum of the partial effects, which would be determined separately,by supposing each of the layers to have been alone heated. Thesame consequence extends to all other problems of the theory ofheat; it is derived from the very nature of equations, and the formof the integrals makes it evident. We see that the heat con-tained in each element of a solid body produces its distinct effect,as if that element had alone been heated, all the others havingnul initial temperature. These separate states are in a mannersuperposed, and unite to form the general system of temperatures.

For this reason the form of the function which represents theinitial state must be regarded as entirely arbitrary. The definiteintegral which enters into the expression of the variable tempera-ture, having the same limits as the heated solid, shows expresslythat we unite all the partial effects due to the initial heating ofeach element.

428. Here we shall terminate this section, which is devotedalmost entirely to analysis. The integrals which we have obtainedare not only general expressions which satisfy the differential equa-tions ; they represent in the most distinct manner the natural effectwhich is the object of the problem. This is the chief condition whichwe have always had in view, and without which the results of in-vestigation would appear to us to be only useless transformations.When this condition is fulfilled, the integral is, properly speaking,the equation of the phenomenon; it expresses clearly the characterand progress of it, in the same manner as the finite equation of aline or curved surface makes known all the properties of thoseforms. To exhibit the solutions, we do not consider one form onlyof the integral; we seek to obtain directly that which is suitableto the problem. Thus it is that the integral which expresses the

SECT. IV.] ELEMENTS OF THE METHOD PURSUED. 451

movement of heat in a sphere of given radius, is very differentfrom that which expresses the movement in a cylindrical body, oreven in a sphere whose radius is supposed infinite. Now each ofthese integrals has a definite form which cannot be replaced byanother. It is necessary to make use of it, if we wish to ascertainthe distribution of heat in the body in question. In general, wecould not introduce any change in the form of onr solutions, with-out making them lose their essential character, which is the repre-sentation of the phenomena.

The different integrals might be derived from each other,since they are co-extensive. But these transformations requirelong calculations, and almost always suppose that the form of theresult is known in advance. We may consider in the first place,bodies whose dimensions are finite, and pass from this problem tothat which relates to ari unbounded solid. We can then substitute adefinite integral for the sum denoted by the symbol 2. Thus it isthat equations (a) and (/3), referred to at the beginning of thissection, depend upon each other. The first becomes the second,when we suppose the radius R infinite. Reciprocally we mayderive from the second equation ($) the solutions relating tobodies of limited dimensions.

In general, we have sought to obtain each result by the shortestway. The chief elements of the method we have' followed arethese:

1st. We consider at the same time the general condition givenby the partial differential equation, and all the special conditionswhich determine the problem completely, and we proceed to formthe analytical expression which satisfies all these conditions.

2nd. We first perceive that this expression contains an infinitenumber of terms, into which unknown constants enter, or thatit is equal to an integral which includes one or more arbitraryfunctions. In the first instance, that is to say, when the generalterm is affected by the symbol 2> we derive from the special con-ditions a definite transcendental equation, whose roots give thevalues of an infinite number of constants.

The second instance obtains when the general term becomes aninfinitely small quantity; the sum of the series is then changedinto a definite integral.

29—2


3rd. We can prove by the fundamental theorems of algebra,or even by the physical nature of the problem, that the transcen-dental equation has all its roots real, in number infinite.

4th. In elementary problems, the general term takes the formof a sine or cosine; the roots of the definite equation are eitherwhole numbers, or real or irrational quantities, each of them in-cluded between two definite limits.

In more complex problems, the general term takes the form ofa function given implicitly by means of a differential equationintegrable or not. However it may be, the roots of the definiteequation exist, they are real, infinite in number. This distinctionof the parts of which the integral must be composed, is veryimportant, since it shews clearly the form of the solution, and thenecessary relation between the coefficients.

5th. It remains only to determine the constants which dependon the initial state; which is done by elimination of the unknownsfrom an infinite number of equations of the first degree. Wemultiply trie equation which relates to the initial state by adifferential factor, and integrate it between defined'limits, whichare most commonly those of the solid in which the movement iseffected.

There are problems in which we have determined the co-efficients by successive integrations, as may be seen in the 'memoirwhose object is the temperature of dwellings. In this case weconsider the exponential integrals, which belong to the initialstate of the infinite solid: it is easy to obtain these integrals1.

It follows from the integrations that all the terms of the secondmember disappear, except only that whose coefficient we wish todetermine. In the value of this coefficient, the denominator be-comes, nul, and we always obtain a definite integral whose limitsare those of the solid, and one of whose factors is the arbitraryfunction which belongs to the initial state. This form of the resultis necessary, since the variable movement, which is the object ofthe problem, is compounded of all those which would have existedseparately, if each point of the solid had alone been heated, andthe temperature of every other point had been nothing.

1 See section 11 of the sketch of this memoir, given by the author in theBulletin des Sciences par la Societe Philomatique, 1818, pp. 1—11. [A. F.]

SECT. IV.J ANALYSIS OF THE PHENOMENON. 453

When we examine carefully the process of integration whichserves to determine the coefficients, we see that it contains acomplete proof, and shews distinctly the nature of the results,so that it is in no way necessary to verify them by other investi-gations.

The most remarkable of the problems which we have hithertopropounded, and the most suitable for shewing the whole of ouranalysis, is that of the movement of heat in a cylindrical body.In other researches, the determination of the coefficients wouldrequire processes of investigation which we do not yet know. Butit must be remarked, that, without determining the values of thecoefficients, we can always acquire an exact knowledge of theproblem, and of the natural course of the phenomenon which isits object; the chief consideration is that of simple movements.

6th. When the expression sought contains a definite integral,the unknown functions arranged under the symbol of integrationare determined, either by the theorems which we have given forthe expression of arbitrary functions in definite integrals, or bya more complex process, several examples of which will be foundin the Second Part.

These theorems can be extended to any number of variables.They belong in some respects to an inverse method of definiteintegration; since they serve to determine under the symbols

and 2 unknown functions which must be such that the result of

integration is a given function.

The same principles are applicable to different other problemsof geometry, of general physics, or of analysis, whether the equa-tions contain finite or infinitely small differences, or whether theycontain both.

The solutions which are obtained by this method are complete,and consist of general integrals. No other integral can be moreextensive. The objections which have been made to this subjectare devoid of all foundation ; it would be superfluous now to discussthem.

7th. We have said that each of these solutions gives the equa-tion proper to the phenomenon, since it represents it distinctly

454* THEORY OF HEAT. [CHAP. IX.

throughout the whole extent of its course, and serves to determinewith facility all its results numerically.

The functions which are obtained by these solutions are thencomposed of a multitude of terms, either- finite or infinitely small:but the form of these expressions is in no degree arbitrary; it isdetermined by the physical character of the phenomenon. Forthis reason, when the value of the function is expressed by a seriesinto which exponentials relative to the time enter, it is ofnecessity that this should be so, since the natural effect whoselaws we seek, is really decomposed into distinct parts, corre-sponding to the different terms of the series. The parts expressso many simple movements compatible with the special conditions ;for each one of these movements, all the temperatures decrease,preserving their primitive ratios. In this composition we oughtnot to see a result of analysis due to the linear form of thedifferential equations, but an actual effect which becomes sensiblein experiments. It appears also in dynamical problems in whichwe consider the causes which destroy motion; but it belongsnecessarily to all problems of the theory of heat, and determinesthe nature of the method which we have followed for the solutionof them.

8th. The mathematical theory of heat includes : first, the exactdefinition of all the elements of the analysis ; next, tl)e differentialequations; lastly, the integrals appropriate to the fundamentalproblems. The equations can be arrived at in several ways; thesame integrals can also be obtained, or other problems solved, byintroducing certain changes in the course of the investigation.We consider that these researches do not constitute a methoddifferent from our own ; but confirm and multiply its results.

9th. It has been objected, to the subject of our analysis, thatthe transcendental equations which determine the exponents havingimaginary roots, it would be necessary to employ the terms whichproceed from them, and which would indicate a periodic characterin part of the phenomenon; but this objection has no foundation,since the equations in question have in fact all their roots real, andno part of the phenomenon can be periodic.

10th. It has been alleged that in order to solve with certaintyproblems of this kind, it is necessary to resort in all cases to a

SECT. IV.J SEPARATE FUNCTIONS. 455

certain form of the integral which was denoted as general; andequation (7) of Art. 398 was propounded under this designa-tion ; but this distinction has no foundation, and the use of asingle integral would only have the effect, in most cases, of com-plicating the investigation unnecessarily. It is moreover evidentthat this integral (7) is derivable from that which we gave in 1807to determine the movement of heat in a ring of definite radius B;it is sufficient to give to R an infinite value.

11th. It has been supposed that the method which consists inexpressing the integral by a succession of exponential terms, andin determining their coefficients by means of the initial state,does not solve the problem of a prism which loses heat unequallyat its two ends; or that, at least, it would be very difficult toverify in this manner the solution derivable from the integral (7)by long calculations. We shall perceive, by a new examination,that our method applies directly to this problem, and that a singleintegration even is sufficient1.

12th. We have developed in series of sines of multiple arcsfunctions which appear to contain only even powers of the variable,cosa? for example. We have expressed by convergent series orby definite integrals separate parts of different functions, or func-tions discontinuous between certain limits, for example that whichmeasures the ordinate of a triangle. Our proofs leave no doubtof the exact truth of these equations.

13th. We find in the works of many geometers results and pro-cesses of calculation analogous to those which we have employed.These are particular cases of a general method, which had not yetbeen formed, and which it became necessary to establish in orderto ascertain even in the most simple problems the mathematicallaws of the distribution of heat. This theory required an analysisappropriate to it, one principal element of which is the analyticalexpression of separate functions, or of parts of functions.

By a separate function, or part of a function, we understand afunction / (oc) which has values existing when the variable x isincluded between given limits, and whose value is always nothing,if the variable is not included between those limits. This func-tion measures the ordinate of a line which includes a finite arc of

1 See the Memoir referred to in note 1, p. 12. [A. F.]


arbitrary form, and coincides with the axis of abscissae in all therest of its course.

This motion is not opposed to the general principles of analysis;we might even find the first traces of it in the writings of DanielBernouilli, of Cauchy, of Lagrange and Euler. It had always beenregarded as manifestly impossible to express in a series of sinesof multiple arcs, or at least in a trigonometric convergent series,a function which has no existing values unless the values of thevariable are included between certain limits, all the other valuesof the function being nul. But this point of analysis is fullycleared up, and it remains incontestable that separate functions,or parts of functions, are exactly expressed by trigonometric con-vergent series, or by definite integrals. We have insisted on thisconsequence from the origin of our researches up to the presenttime, since we are not concerned here with an abstract and isolatedproblem, but with a primary consideration intimately connectedwith the most useful and extensive considerations. Nothing hasappeared to us more suitable than geometrical constructions todemonstrate the truth of these new results, and to render intelli-gible the forms which analysis employs fGr their expression.

14th. The principles which have served to establish for us theanalytical theory of heat, apply directly to the investigation of themovement of waves in fluids, a part of which has been agitated.They aid also the investigation of the vibrations of elastic laminae,of stretched flexible surfaces, of plane elastic surfaces of very greatdimensions, and apply in general to problems which depend uponthe theory of elasticity. The property of the solutions which wederive from these principles is to render the numerical applicationseasy, and to offer distinct and intelligible results, which reallydetermine the object of the problem, without making that know-ledge depend upon integrations or eliminations which cannot beeffected. We regard as superfluous every transformation of theresults of analysis which does not satisfy this primary condition,

429. 1st. We shall now make some remarks on the differen-tial equations of the movement of heat.

If two molecules of the same body are extremely near, and areat unequal temperatures, that which is the most heated communicates

SECT. IV.] FOKMATION OF EQUATIONS OF MOVEMENT. 4 5 7

directly to the other during one instant a certain quantity of heat;which quantity is proportional to the extremely small difference ofthe temperatures: that is to say, if that difference became double,triple, quadruple, and all other conditions remained the same, theheat communicated would be double, triple, quadruple.

This proposition expresses a general and constant fact, whichis sufficient to serve as the foundation of the mathematical theory.The mode of transmission is then known with certainty, inde-pendently of every hypothesis on the nature of the cause, andcannot be looked at from two different points of view. It isevident that the direct transfer is effected in all directions, andthat it has no existence in fluids or liquids which are not diather-manous, except between extremely near molecules.

The general equations of the movement of heat, in theinterior of solids of any dimensions, and at the surface of thesebodies, are necessary consequences of the foregoing proposition.They are rigorously derived from it, as we have proved in ourfirst Memoirs in 1807, and we easily obtain these equations bymeans of lemmas, whose proof is not less exact than that of theelementary propositions of mechanics.

These equations are again derived from the same proposition,by determining by means of integrations the whole quantity ofheat which one molecule receives from those which surround it.This investigation is subject to no difficulty. The lemmas inquestion take the place of the integrations, since they give directlythe expression of the flow, that is to say of the quantity of heat,which crosses any section. Both calculations ought evidently tolead to the same result; and since there is no difference in theprinciple, there cannot be any difference in the consequences.

2nd. We gave in 1811 the general equation relative to thesurface. It has not been deduced from particular cases, as hasbeen supposed without any foundation, and it could not be; theproposition which it expresses is not of a nature to be discoveredby way of induction; we cannot ascertain it for certain bodies andignore it for others; it is necessary for all, in order that the stateof the surface may not suffer in a definite time an infinite change.In our Memoir we have omitted the details of the proof, since


they consist solely in the application of known propositions. Itwas sufficient in this work to give the principle and the result, aswe have done in Article 15 of the Memoir cited. From the samecondition also the general equation in question is derived by deter-mining the whole quantity of heat which each molecule situatedat the surface receives and communicates. These very complexcalculations make no change in the nature of the proof.

In the investigation of the differential equation of the move-ment of heat, the mass may be supposed to be not homogeneous,and it is very easy to derive the equation from the analyticalexpression of the flow; it is sufficient to leave the coefficient whichmeasures the conducibility under the sign of differentiation.

3rd. NewTton was the first to consider the law of cooling ofbodies in air; that which he has adopted for the case in which theair is carried away with constant velocity accords more closelywith observation as the difference of temperatures becomes less;it would exactly hold if that difference were infinitely small.

Amontons has made a remarkable experiment on the establish-ment of heat in a prism whose extremity is submitted to a definitetemperature. The logarithmic law of the decrease of the tempera-tures in the prism was given for the first tirne by Lambert, of theAcademy of Berlin. Biot and Rumford have confirmed this lawby experiment1.

1 Newton, at the end of his Scala graduum caloris et frigoris, PhilosophicalTransactions, April 1701, or Opuscula ed. Castillioneus, Vol. n. implies that whena plate of iron cools in a current of air flowing uniformly at constant temperature,equal quantities of air come in contact with the metal in equal times and carryoff quantities of heat proportional to the excess of the temperature of the ironover that of the air; whence it maybe inferred that the excess temperatures ofthe iron form a geometrical progression at times which are in arithmetic progres-sion, as he has stated. By placing various substances on the heated iron, heobtained their melting points as the metal cooled.

Amontons, Memoircs de VAcademic [1703], Paris, 1705, pp. 205—6, in hisMemarques sur la Table de dcgrca de Chaleur extraite des Transactions Philosophi-ques 1701, states that he obtained the melting points of the substances experimentedon by Newton by placing them at appropriate points along an iron bar, heated towhiteness at one end; but he has made an erroneous assumption as to the lawof decrease of temperature along the bar.

Lambert, Pyromctrie, Berlin, 1779, pp. 185—6, combining Newton's calculatedtemperatures with Amontons' measured distances, detected the exponential law

SECT. IV.] LAW OF THE FLOW OF HEAT. 459

To discover the differential equations of the variable movementof heat, even in the most elementary case, as that of a cylindricalprism of very small radius, it was necessary to know the mathe-matical expression of the quantity of heat which traverses anextremely short part of the prism. This quantity is not simplyproportional to the difference of the temperatures of the twosections which bound the layer. It is proved in the most rigorousmanner that it is also in the inverse ratio of the thickness of thelayer, that is to say, that if two layers of the same prism were un-equally thick, and if in the first the difference of the temperatures ofthe two bases was the same as in the second, the quantities of heattraversing the layers during the same instant would be in the inverseratio of the thicknesses. The preceding lemma applies not only tolayers whose thickness is infinitely small; it applies to prisms ofany length. This notion of the flow is fundamental; in so far aswe have not acquired it, we cannot form an exact idea of thephenomenon and of the equation which expresses it.

It is evident that the instantaneous increase of the tempera-

of temperatures in a long bar heated at one end. Lambert's work contains amost complete account of the progress of thermal measurement up to that time.

Biot, Journal des Mines, Paris, 1804, xvn. pp. 203—224. Rumford, Memoiresde VInstitut, Sciences Math, et Phys. Tome vi. Paris, 1805, pp. 106—122.

Ericsson, Nature, Vol. vi. pp. 106—8, describes some experiments on coolingin vacuo which for a limited range of excess temperature, 10° to 100° Fah. shewa very close approach to Newton's law of cooling in a current of air. Theseexperiments are insufficient to discredit the law of cooling in vacuo derived byM. M. Dulong and Petit (Journal Poly technique, Tome xi. or Ann. de Ch. etde Ph. 1817, Tome vn.) from their carefully devised and more extensive rangeof experiments. But other experiments made by Ericsson with an ingeniouslycontrived calorimeter (Nature, Vol. v. pp. 505—7) on the emissive power of molteniron, seem to shew that the law of Dulong and Petit, for cooling in vacuo, isvery far from being applicable to masses at exceedingly high temperatures givingoff heat in free air, though their law for such conditions is reducible to the formerlaw.

Fourier has published some remarks on Newton's law of cooling in hisQuestions sur la theorie physique de la Chaleur rayonnante, Ann. de Chimie et dePhysique, 1817, Tome vi. p. 298. He distinguishes between the surface conductionand radiation to free air.

Newton's original statement in the Scala graduum is " Calor quern ferrumcalefactum corporibus frigidis sibi contiguis dato tempore communicat, hoc estCalor, quern ferrum dato tempore amittit, est ut Calor totus ferri." This supposesthe iron to be perfectly conducible, and the surrounding masses to be at zerotemperature. It can only be interpreted by his subsequent explanation, as above.

[A. R]


ture of a point is proportional to the excess of the quantity of heatwhich that point receives over the quantity which it has lost, andthat a partial differential equation must express this result: butthe problem does not consist in enunciating this proposition whichis the mere fact; it consists in actually forming the differentialequation, which requires that we should consider the fact in itselements. If instead of employing the exact expression of theflow of heat, we omit the denominator of this expression, wethereby introduce a difficulty which is nowise inherent in theproblem; there is no mathematical theory which would not offersimilar difficulties, if we began by altering the principle of theproofs. Not only are we thus unable to form a differential equa-tion; but there is nothing more opposite to an equation than aproposition of this kind, in which we should be expressing theequality of quantities which could not be compared. To avoidthis error, it is sufficient to give some attention to the demon-stration and the consequences of the foregoing lemma (Art. 65,66, 67, and Art. 75).

4th. With respect to the ideas from which we have deducedfor the first time the differential equations, they are those whichphysicists have always admitted. We do not know that anyonehas been able to imagine the movement of heat as being producedin the interior of bodies by the simple contact of the surfaceswhich separate the different parts. For ourselves such a propositionwould appear to be void of all intelligible meaning. A surface ofcontact cannot be the subject of any physical quality; it is neitherheated, nor coloured, nor heavy. I t is evident that when onepart of a body gives its heat to another there are an infinityof material points of the first which act on an infinity of points ofthe second. It need only be added that in the interior of opaquematerial, points whose distance is not very small cannot commu-nicate their heat directly; that which they send out is interceptedby the intermediate molecules. The layers in contact are the onlyones which communicate their heat directly, when the thicknessof the layers equals or exceeds the distance which the heat sentfrom a point passes over before being entirely absorbed. There isno direct action except between material points extremely near,and it is for this reason that the expression for the flow has theform which we assign to it. The flow then results from an infinite

SECT. IV.] FLOW OUTWARD AND INTERNAL. 4 6 1

multitude of actions whose effects are added; but it is not fromthis cause that its value during unit of time is a finite andmeasurable magnitude, even although it be determined only byan extremely small difference between the temperatures.

When a heated body loses its heat in an elastic medium, or ina space free from air bounded by a solid envelope, the value of theoutward flow is assuredly an integral; it again is due to the actionof an infinity of material points, very near to the surface, and wehave proved formerly that this concourse determines the law ofthe external radiation1. But the quantity of heat emitted duringthe unit of time would be infinitely small, if the difference of thetemperatures had not a finite value.

In the interior of masses the conductive power is incomparablygreater than that which is exerted at the surface. This property,whatever be the cause of it, is most distinctly perceived by us,since, when the prism has arrived at its constant state, thequantity of heat which crosses a section during the unit of timeexactly balances that which is lost through the whole part of theheated surface, situated beyond that section, whose temperaturesexceed that of the medium by a finite magnitude. When we takeno account of this primary fact, and omit the divisor in theexpression for the flow, it is quite impossible to form the differen-tial equation, even for the simplest case; a fortiori, we should bestopped in the investigation of the general equations.

5th. Farther, it is necessary to know what is the influence ofthe dimensions of the section of the prism on the values of theacquired temperatures. Even although the problem is only thatof the linear movement, and all points of a section are regardedas having the same temperature, it does not follow that we candisregard the dimensions of the section, and extend to other prismsthe consequences which belong to one prism only. The exactequation cannot be formed without expressing the relationbetween the extent of the section and the effect produced at theextremity of the prism.

We shall not develope further the examination of the principleswhich have led us to the knowledge of the differential equations;

1 Memoires de VAcademic des Sciences, Tome v. pp. 204—8. Communicatedin 1811. [A. F.]


we need only add that to obtain a profound conviction of the use-fulness of these principles it is necessary to consider also variousdifficult problems; for example, that which we are about to in-dicate, and whose solution is wanting to our theory, as we havelong since remarked. This problem consists in forming the differ-ential equations, which express the distribution of heat in fluidsin motion, when all the molecules are displaced by any forces,combined with the changes of temperature. The equations whichwe gave in the course of the year 1820 belong to general hydro-dynamics; they complete this branch of analytical mechanics1.

430. Different bodies enjoy very unequally the property whichphysicists have called conductibility or conducibility, that is to say,the faculty of admitting heat, or of propagating it in the interiorof their masses. We have not changed these names, though they

1 See Memoires de VAcademie des Sciences, Tome xn. Paris, 1833, pp. 515—530.In addition to the three ordinary equations of motion of an incompressible

fluid, and the equation of continuity referred to rectangular axes in direction ofwhich the velocities of a molecule passing the point x, y, z at time t are u, v, iv,its temperature being 6, Fourier has obtained the equation

in which K is the conductivity and C the specific heat per unit volume, asfollows.

Into the parallelopiped whose opposite corners are (x, y, z), (x + Ax, y + Ay, z + Az),the quantity of heat which would flow by conduction across the lower face Ax Ay,

if the fluid were at rest, would be -K—Ax Ay At in time At, and the gain by

convection + Cw Ax Ay At; there is a corresponding loss at the upper face Ax Ay;

hence the whole gain is, negatively, the variation of (-K --+ Cwd) Ax Ay At with

respect to z, that is to say, the gain is equal to (K-p2-C,(w6)j Ax Ay Az At.

Two similar expressions denote the gains in direction of y and z; the sum of the7/3

three is equal to C j-At Ax Ay A:, which is the gain in the volume Ax Ay Az

in time At: whence the above equation.The coefficients K and C vary with the temperature and pressure but are

usually treated as constant. The density, even for fluids denominated incom-pressible, is subject to a small temperature variation.

It may be noticed that when the velocities u, v, w are nul, the equationreduces to the equation for flow of heat in a solid.

It may also be remarked that when K is so small as to be negligible, theequation has the same form as the equation of continuity. [A. F. ]

SECT. IV.] PENETRABILITY AND PERMEABILITY. 4G3

do not appear to us to be exact. Each of them, the first especially,would rather express, according to all analogy, the faculty of beingconducted than that of conducting.

Heat penetrates the surface of different substances with moreor less facility, whether it be to enter or to escape, and bodies areunequally permeable to this element, that is to'say, it is propagatedin them with more or less facility, in passing from one interiormolecule to another. We think these two distinct propertiesmight be denoted by the names penetrability and permeability1.

Above all it must not be lost sight of that the penetrability ofa surface depends upon two different qualities: one relative to theexternal medium, which expresses the facility of communication bycontact; the other consists in the property of emitting or admit-ting radiant heat. With regard to the specific permeability, it isproper to each substance and independent of the state of thesurface. For the rest, precise definitions are the true foundationof theory, but names have not, in the matter of our subject, thesame degree of importance.

431. The last remark cannot be applied to notations, whichcontribute very much to the progress of the science of the Calculus.These ought only to be proposed with reserve, and not admittedbut after long examination. That which we have employed re-duces itself to indicating the limits of the integral above and below

the sign of integration I; writing immediately after this sign the

differential of the quantity which varies between these limits.

We have availed ourselves also of the sign 2 to express thesum of an indefinite number of terms derived from one generalterm in which the index i is made to vary. We attach this indexif necessary to the sign, and write the first value of i below, andthe last above. Habitual use of this notation convinces us of

1 The coefficients of penetrability and permeability, or of exterior and interiorconduction (h, K), were determined in the first instance by Fourier, for the caseof cast iron, by experiments on the permanent temperatures of a ring and on the

varying temperatures of a sphere. The value of - by the method of Art. 110,

and the value of h by that of Art. 297. Mem. de VAcad. d. Sc. Tome v. pp.

165, 220, 228. [A. F.]


the usefulness of it, especially when the analysis consists of de-finite integrals, and the limits of the integrals are themselves theobject of investigation.

432. The chief results of our theory are the differential equa-tions of the movement of heat in solid or liqiiid bodies, and thegeneral equation which relates to the surface. The truth of theseequations is not founded on any physical explanation of the effectsof heat. In whatever manner we please to imagine the nature ofthis element, whether we regard it as a distinct material thingwhich passes from one p&rt of space to another, or whether wemake heat consist simply in the transfer of motion, we shall alwaysarrive at the same equations, since the hypothesis which we formmust represent the general and simple facts from which themathematical laws are derived.

The quantity of heat transmitted by two molecules whosetemperatures are unequal, depends on the difference of thesetemperatures. If the difference is infinitely small it is certainthat the heat communicated is proportional to that difference; allexperiment concurs in rigorously proving this proposition. Nowin order to establish the differential equations in question, weconsider only the reciprocal action of molecules infinitely near.There is therefore no uncertainty about the form of the equationswhich relate to the interior of the mass.

The equation relative to the surface expresses, as we have said,that the flow of the heat, in the direction of the normal at theboundary of the solid, must have the same value, whether we cal-culate the mutual action of the molecules of the solid, or whetherwe consider the action which the medium exerts upon the envelope.The analytical expression of the former value is very simple andis exactly known; as to the latter value, it is sensibly proportionalto the temperature of the surface, when the excess of this tempera-ture over that of the medium is a sufficiently small quantity. Inother cases the second value must be regarded as given by a seriesof observations; it depends on the surface, on the pressure andon the nature of the medium ; this observed value ought to formthe second member of the equation relative to the surface.

In several important problems, the equation last named is re-

SECT. IV.] THREE SPECIFIC COEFFICIENTS. 465

placed by a given condition, which expresses the state of thesurface^ whether constant, variable or periodic.

433. The differential equations of the movement of heat aremathematical consequences analogous to the general equations ofequilibrium and of motion, and are derived like them from themost constant natural facts.

The coefficients c, h, h, which enter into these equations, mustbe considered, in general, as variable magnitudes, which dependon the temperature or on the state of the body. But in the appli-cation to the natural problems which interest us most, we mayassign to these coefficients values sensibly constant.

The first coefficient c varies very slowly, according as the tem-perature rises. These changes are almost insensible in an intervalof about thirty degrees. A series of valuable observations, due toProfessors Dulong and Petit, indicates that the value of the specificcapacity increases very slowly with the temperature.

The coefficient h which measures the penetrability of the sur-face is most variable, and relates to a very composite state. Itexpresses the quantity of heat communicated to the medium,whether by radiation, or by contact. The rigorous calculation ofthis quantity would depend therefore on the problem of the move-ment of heat in liquid or aeriform media. But when the excessof temperature is a sufficiently small quantity, the observationsprove that the value of the coefficient may be regarded as constant.In other cases, it is easy to derive from known experiments acorrection which makes the result sufficiently exact.

It cannot be doubted that the coefficient k, the measure of thepermeability, is subject to sensible variations; but on this impor-tant subject no series of experiments has yet been made suitablefor informing us how the facility of conduction of heat changes withthe temperature1 and with the pressure. We see, from the obser-vations, that this quality may be regarded as constant throughouta very great part of the thermometric scale. But the same obser-vations would lead us to believe that the value of the coefficientin question, is very much more changed by increments of tempera-ture than the value of the specific capacity.

Lastly, the dilatability of solids, or their tendency to increase1 Eeference is given to Forbes' experiments in the note, p. 84. [A. F.]

F. H. 30


in volume, is not the same at all temperatures: but in the problemswhich we have discussed, these changes cannot sensibly alter theprecision of the results. In general, in the study of the grandnatural phenomena which depend on the distribution of heat, werely on regarding the values of the coefficients as constant. It isnecessary, first, to consider the consequences of the theory fromthis point of view. Careful comparison of the results with thoseof very exact experiments will then shew what corrections must beemployed, and to the theoretical researches will be given a furtherextension, according as the observations become more numerousand more exact. We shall then ascertain what are the causeswhich modify the movement of heat in the interior of bodies,and the theory will acquire a perfection which it would be im-possible to give to it at present.

Luminous heat, or that which accompanies the rays of lightemitted by incandescent bodies, penetrates transparent solids andliquids, and is gradually absorbed within them after traversing aninterval of sensible magnitude. It could not therefore be supposedin the examination of these problems, that the direct impressionsof heat are conveyed only to an extremely small distance. Whenthis distance has a finite value, the differential equations take adifferent form; but this part of the theory would offer no usefulapplications unless it were based upon experimental knowledgewhich we have not yet acquired.

The experiments indicate that, at moderate temperatures, avery feeble portion of the obscure heat enjoys the same property asthe luminous heat; it is very likely that the distance, to which isconveyed the impression of heat which penetrates solids, is notwholly insensible, and that it is only very small: but this occasionsno appreciable difference in the results of theory; or at least thedifference has hitherto escaped all observation.

The Analytical Theory of Heat - Jean Baptiste Joseph Fourier

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