The Messenger of Mathematics v17 1000034177

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-^^

^Ic^

^

THE

MESSENGER OF

MATHEMATICS.

EDITED BT

J.

W.

L.

GLAISHER,

So.D.,

P.E.8.,

niLIiOW

or TRINITT

OOLLiail,

OAKBBIDSH.

VOL. XVII.

[Mat,

1887

Apkil,

1888.]

MAOMILLAN

AND

00.

Honlron

anlr

Cambtfiige.

1888.



irr

7;

j/^^z .^

0^

lii'h

^fjc,

G^

CAMBRIDGE)

:

PRINTED

BT

W.

MBTOALPK

AND

SON,

TRINITY

STREET

AND ROSE

CRESCENT.



CONTENTS.

ARITHMETIC,

ALGEBRA,

AND

TRIGONOMETRY.

On

the

order

of

proof

of

the

principal

equations

of

spherical

trigonometry.

By

M.

Jenkins

.------

Note

on

a

theorem

in

Higher

Algebra.

By

H.

G.

Dawsok

On

a

theorem

of

Prof.

Klein's

relating

to

symmetric

matrices.

By

A.

BnCHHEIM

.......

A

new

method

for

the

graphical representation

of

complex quantities.

By

J.

Brill

---.---

Note

on

the

anharmonic

ratio

equation.

By

Prof.

Catlet

-

Note

on

the

multiplication

of

nonions.

By

G. G.

Morricb

.

An

extension

of

a

certain

theorem

in

inequalities.

By

L.

J.

Roobrs

PAGE

30

69

79

80

95

104

145

GEOMETRY

OP

TWO

AND

THREE

DIMENSIONS.

System

of

equations

for

three circles

which

cut

each other

at

given angles.

By

Prof. Catlet

.......

18

On

plane

cubics

which

inflect

on

crossing

their

asymptotes.

By

F,

Morlbt

61

Note

on

certain

theorems

relating

to

the

polar

circle

of

a

triangle

and

Feuerbach's

theorem

on

the

nine-point

circle.

By

S. Roberts

-

57

On

the

system

of

three circles which

cut

each

other

at

given angles

and

hare their centres in

a

line.

By

Prof.

Catlet

- - -

60

On

systems

of

rays.

By

Prof. Catlet

-

-

- - -

78

Note

on

the

two

relations

connecting

the

distances

of

four

points

on

a

circle.

By

Prof,

Catlet

-

-

-

- - -

94

The

cosine

orthocentres

of

a

triangle

and

a

cubic

through

them.

By

R.

Tttokbr

--.-.--

Geometry

on a quadric

surface.

By

Prof.

Mathews

-

97

-

151



IV

CONTENTS,

DIFFERENTIAL

AND INTEGRAL CALCULUS

AND

DIFFERENTIAL

EQUATIONS.

.

PAGE

Note

on

the

Legendrian

coefficients

of

the

second

kind.

By

Prof. Cayley

-

21

The

transformation

of

multiple

surface

integrals

nto

multiple

line

integrals.

ByJ.

Larmor --------23

Depression

of

differential

equations.By

Lt.-Col.

Allan

Cunningham

-

118

nomographic

inyariants

and

quotient

deriyatiyes.

By

A.

R.

Forsyth

-

154

THEORY OF ELLIPTIC FUNCTIONS.

On the

tn^nsformation

and

derelopments

of the

twelve

elliptic

unctions

and

the

four

Zeta functions.

By

J.

W.

L.

Glaishhr

-

-

-

1

On the

second

solution

of

the

differential

equation

of

the

hypergeometric

series,

nd the

series

for

K'f

E',

c.,

in

Elliptic

unctions.

By

Brof.

W. Woolsby

Johnson

-------

35

Expressions

for

8

(a;)

s

a

definite

integral.

By

J. W. L. Glaishbr

-

152

APPLIED MATHEMATICS.

Yortices

in

a

compressible

luid.

By

C. Chrbb

....

105



n



MESSENGER

OF MATHEMATICS.

ON

THE

TRANSFORMATION

AND

DEVELOP-ENTS

OF

THE

TWELVE

ELLIPTIC

FUNCTIONS

AND

THE

FOUR ZETA

FUNCTIONS.

By

J,

W.

L.

Olaisher.

The

principalbject

f

this

paper

is

to

give

the transfor-ations

of

the

elliptic

nd

Zeta

functions,

which

are

due

to

the

change

of

q

into

-

q^

^

and

q^

;

and

also the firstfew

terms

of

the

expansions

of these

functions

in

ascending

powers

of the

arguments.

I omit the

demonstrations,iving

only

the

resultsin

a

form

convenient

for

reference.

This

paper

(in

so

far

as

it

relates

to

the

elliptic

unctions)

may

be

regarded

as

a

continuation

of

two

papers

^^

On

[Elliptic

unctions,

hich

were

published

n

vol.

XI.

of

the

Messenger,

pp.81-95,

120-138).

In

17 19

I have

given

tables

of the

values of

the

elliptic

unctions

when the

argument

is

increased

by Jf,

iK*

or

K-\-iK\

and

also the values of

the

functions

for certain

special

alues of the

argument.

I

have

found these tables of

such

constant

use

in

working

ith

the

twelve

elliptic

unctions

that

I

have

been

tempted

to

include

them in this

paper.

The

firsttable

in

17

was

given

in vol.

XI.,

p.

88,

but

the

functions

were

there

arranged

in

an

inconvenient order.

For the sake of

completeness

he

g'-series

or

the

elliptic

and

Zeta

functions

are

also

given

(

2,

3).

Notation^

1.

The letters

p

and

u

are

used

to

denote

2Jr

,

^Kx

and

IT IT

respectively

so

that

u

=

px.

The

argument

x

is

supposed

o

be

independent

f

k.

The

lettersh and Ji

are

used

to

denote

k^

and

A'*

respectively,

VOL, XVII.

B



2

MB.

QLAISHER,

TBANSFQBMATION

AND

DEVELOPMENTS

The

q^series

for

Icp

nu,

c.,

and

pZ{u)^

2.

2.

The

j-series

or the

twelve

elliptic

unctions

are

:*

hp

snw

=

2*

^

^n-i

sin

(2n

1)

a?,

kp

cnM=

2

y^7-5n=i

os

(2 i

1)

a?,

pdnttsa

27

.

,n

co82yia?;

p

nsM=

-:;

+

2*

T-^-i =i

in

(2n

1)

x.

'^

sina;

*

1

-

^*^

^ ' '

p

dsw

=

-:

2

,

^

n-i

sin

(2n 1)

a?,

^

sina?

*

1+2'

'

k'p

ncu^

2r

(-p/^'-i

os

(2n

1)

a?,

^

cosa?

'

^

'

1

+

2

Ap

cdtt=

Sr

(-) -

j^^,

os

(2n

1)

x,

Mp

8d

u

=

Sr (-r

^^

n

(2n

1)

x,

A;

nd

tt

=

1

-

2^

(-) -*

j-^

os

2nar

;

Meittnger,

oI,

xvi.,

pp.

187,

188.



OF

THE TWELVB

ELLIPTIC

AND

FOUR

ZETA

FUNCTIONS.

3

and

those

for the

four

Zeta

functions

are

;*

pz

(u)

sr

Yif^

^

'^'^^

i

to

th

2Van /brmatu M

of

kp

snu,

c.,

3,

4.

3.

The

changes

n the functions

hp

sn

u,

c irhich

are

dae

The

changes

in

the

functions

which

are

due

to

the

change

of

2

into

^

and

^

are

shown

in

the

table

on

the

next-

page:

t

Messenger

f

yoI.

xv., p.

146.

b2



4

MR.

GLAISHER,

TRANSFORMATION

AND

DEVELOPMENTS



OF THE TWELVE

ELLIPTIC

AND

POUR

ZETA

PUNCTIONS.

5

The

quantities

n'^u

h'

which

occur

in

the

firstcolumn

of

results

may

be

expressed

n the

forms

(l *')(cn'iu *'sn'iM),

and

the

quantities\ 1c)

(\^h

snV)

which

occur

in

the

second column

may

be

expressed

n the forms

dn'tf

k

cn'tf.

4.

In

the

following

ases

the

transformed

quantities

are

expressible

ery

simply

n

terms

of

sums or

differences

of

elliptic

r

Zeta functions.

Transformationsf

pZJ(u)^

5.

5.

The

changes

in

pZ{u) P^,{^)t

c

dae

to

the

change

of

2

into

j*

are :

?.

-?.

pZ( ),

pZ^u),

pZM,

pz,{u\

pZ,{u),

pZ{u),



6

MR.

GLAISHER,

TRANSFORMATION

AND

DEVELOPMENTS

The

changes

hich

are

dne to

the

change

of

q

into

q*

or

q^

are

shown in

the

following

able

:

The

first

two

quantities

n

the

second

column

may

be

expressed

n the forms

\{pZ{^u)^.pZl\u)],

\\pZSMu)-^pZi\u)}

respectively

and

the second and third

quantities

n

the

third

column

may

be

expressed

n the

forms

pZ[u)-^pZ,{u),

Z^{u)+pZJ,u)

respectively.

Transformations

f

sum,

c.,

6.

6.

It

seems

desirable

to

give

also

the

transformations

of the functions

snte,

cnu,

c.

The

.changes

ue

to

the

change

of

q

into

q^x^x



OF

THE

TWELVE

ELLIPTIC

AND

POUR ZETA FUNCTIONS,

7

The

changes

ue

to

the

change

of

q

into

^

and

^

are :

snu

cnu

dnu

nsu

dsu

csu

dcu

ncu

scu

cdu

sdu

niu

^

'^

dnû

dnû

dnû

1

dn^M

l

+

k'

snûcn^t^

1

cn'^M-f-

:'8n'^^

l+k'

snî^cQû

1

cn'^M

k'

snû

cn'^M

+

i'sn'^M

cn^^w

Aj'sn'^M

dn^M

cn'û

k'

sn'^M

( +*')::::?

nûcn^t^

cn*iu

k'

sn^'û

cn'^M

k'

sn'^M

cn'^M

+

i'

sn*^M

a+^')rrf

sn

î

n

iu

cn*iw+

A;'Bn*^w

dnjû

cn^^M

+

k'

sn*û

1

-

A

snû



8

MR.

QLAISMER,

TRANSFORMATION AND

DEVELOPMENTS

Transformations

fZ{u)y

c.,

7.

7.

It

18

UDDecessaiy

to

give

the

tranaforinations

of

-2^(m),

as

they

are

deducible

at

sight

from

those

of

pZJ^u)

given

in

5.

To deduce the

transformations of

Z,{u)

from

those

of

pZ,{u)

it

suffices

to

replace,

n the transformed

results,

p

by

p

in

the

g

column,

1

2

Transformationsf

p,

A:p,

'p,AA'p,

8.

8.

In

connexion

with the

preceding

ystems

of

formulas

it

is

convenient

to

give

the

transformations of

p,

kpy

c.|

which

are

as

follows

:

Transformations

f

sna;,

c.,^{^),

c.,

9.

9.

The transformations of the sixteen functions

snar,

c.

Z[x)^

c.,

are

easily

deducible

from those

of

sum,

c.,

Z(w),

c.,

in

6

and

7

by simply

multiplying

he

trans-ormed

arguments

in

the columns

headed

g,

g^

and

g*

by



OP

THE

TWELVE

ELLIPTIC

AND FOUR

ZETA

FUNCTIONS.

9

For

example,

by

the

change

of

q

into

-;,

q*

and

^S

sn^

becomes

respectively.

The

transformations

xpressed

s

equatwnSy

10.

10.

By

the

change

of

q

into

^q

the

modalas

k

is

converted into

p

and

JTinto

k'K.

By

the

changes

f

q

into

^

and

of

2

into

^

the modulus

X;

is

converted

into

X

and

7^

and

K is

converted

into A

and

F,

where

Denoting

and

by

px

^'^^

Py

^^^ ^^^

^7

^

and

Wj

the

transformations

given

in

the

last

two

sections

may

be

expressed

s

equations

n

the

form

:

ikp

sn

f

A'w,

77

)

=

ilck'p

d

u^

c.,

,

_.

--,

sn^Mcniw

Xpx8n(t;,

)=^A;p

^^^^

,

c.,

7P

en

[w.

7)

=

2A:*p ; /

^ ,

c.,

sn(x/|)=*'sd(|),

c.,

X

X

sn

;

j7

en

sn

(a:,)

=

(1

+

A:')

,

c.,

Bn(ar,7)=

^

C.



10

MR.

aLAISHER,

TRANSFORMATION

AND

DEVELOPMENTS

Expansions

f

the

elliptic

unctions

n

powers

of

x^%\\.

11.

The

expansions

of the

twelve

elliptic

unctions

in

ascending

owers

of

x

are

as

follows

:

8na?

=

;c-(l

+

A)|^

(l

+

14A

+

A')^

a;'

-

(1

+

135A

+

1

35A'

+

A')

y7

+

c.,

cna:

=

1

-

1^

(1

+

4A)

^

-

(1

+

44*

+

16A')

1^

c.,

dna

=

l-A^j

A(A

+

4)|^j-A(A'

44A4l6)^

c.;

n8a

=

^

+

i(l

+

A)a

+

^(7-22A

+

7A')|^

+

tk

(31

ISA

-

15A'

+

31A')

jj

+

c,,

dsa;

=

i

-

J

(A

A')

+

sV

(7A'

22AA'

+

TA )

J^.

-

X

Jy

(31A

15A'A'

-

15AA'

-

3lA )

^j

c.,

C8

=

^

i

(1

+

A')

;

+

j'ty

7

22A'

+

7A )

^j

-

Ti,

(31

15A' -

15A

+

31A'')

+

*c-

J

dca:

=

1

+

A'

^

+

A'

(A'

4)

|^,

A'

(A

+

44A'

+

16)

|^,

c.,

nca;

=

1

+

^

+

(1

+

4A')

2l

+

(1

+

44A'

+

ICA )

^

+ c.,

Bca

=

+

(1

+

A')^

{1

+

14A'

+

A'')^j

+

(1

+

135A'

+

1

35A'

+

A )

^j

fec.

;

.-

cd

=

1

-

A'

Jy

+

A'

(A'-

4A)

Jy

A'

(A -44AA'+

16A')

J-j

c.,

sdar

=

+

(A A')

^j

(A'

14AA'

+

A )

|^

4

(A*

135A'A'

+

135AA

-

A )

,

+

c.,

nda

=

1

+

A

^j

A

(A

4A')

1^

A

(A*

44AA'

+

16A )

1^

c.



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12

MR.

GLAISHEfi,

TRANSFORMATION

AND

DEVELOPMENTS

gz,aj

=

- ;i'

{^

2

^i

2X1

+

A')^j

2X2+13A'+2r)f^

c.l

,

gz,:p

=

AA'|2

j

+2'(A-A')|^j

2*(2A'-13AA'+2r)

c.|

14. The other

Zeta

functions

iz^e,

c.,

differ from

gz cc

only

by

multiples

f

a?,

so

that

with

the

sole

exception

f

the

term

involving

the

expansions

of these functions

are

the

same as

those of

gz/c.

The

accompanying

able

gives

the

value of the

term

involving

for

each

of

the

twelve

functions

iz,a;,

z^,

ez;r.

The

corresponding

erms

in

the

expansions

f

Z^

{x)

and

Z

{x)

are

shown in the

next

table.



OF THE TWELVE

ELLIPTIC AND POUR

ZETA FUNCTIONS.

13

The fanctions

^{x)

are

the

functions

so

desig^nated

n

vol.

XV.

pp.

92-102.

The

quantities

and

H' denote

J

(7+

G

+

JS)

and

J

(/'

+

ff

'

+

')

respectively.

15.

It

will

be

noticed

that

the

series

for

ffza;,

gz,ar,gz,x

contain

A,

A',

hh'

respectively

s

factors.

The

only

other

series

which contain

a

factor

common

to

all

the

terms

are

those for iz^

and

ez^a;,

neither of

which

contains

a

term

in

x.

The three series

which contain

no

term

in

x

are

:

izur=-

aJ2J-2 (1

A)^

2*(2

+13A

+2A )

^j- c.l

ez^=-

*'J2^

2-(l+A')|^+2^(2

13A'

+2A )|^j+i

gz,a?=

AA'

|2

^4

'{A-A')|'j

2*(2A -

13AA'+2A' )

|^+ c.l

16.

The

values of

Z^(x),

when

x

is

small,

are

often

required

in

verifying

formulae

and for

other

purposes.

I therefore

add

for

the sake of

reference

the

expansions

of

Z,

{x)

as

far

as

the

terms

involving

^

:

Z{x)=

-^a?-iAa:',

Increase

of

the

argument

by

K^

iK\

K-\-

{K\

17.

17.

In

working

with

the

twelve

elliptic

unctions

it

is

convenient

to

have for

reference

in

a

tabular

form the

complete

system

of

changes

which

ar^

produced

in

the



14

MR.

GLAISHER,

TRANSFORMATION

AND

DEVELOPMENTS

functions

by

the increase of

the

argument

by

K^

iK and

These

changes

re

shown

in the

following

able

:

The

next

table,

hich

is

deducible

at

once

from

the

last,

gives

the

correspondinghanges

of

A;sna;,

A;

en

a;,

c.



OF THE

TWELVE

ELLIPTIC

AND

FOUR

ZETA

FUNCTIONS.

15

These twelve

fanctions form

a

group

complete

n

itself,

iz.

each

function

is

transformed

into

another

member

of the

group

multiplied

y

1

or

i.

Valttes

of

the

elliptic

unetions

hen

the

argument

is

OjKjiK'jK+iK'y

18.

18.

The

following

table

gives

the

values of the

twelve

elliptic

unctions for the

arguments

0,

Ky iK',

K+

iK*.

The letter

a

denotes

zero

and A

infinity,

ut the

following

more

precise

ignifications

ay

be

attached

to

these

letters^

viz.

the

arguments

0,

K^ iK'^

Jr+

iK'

may

be

regardedas

denoting

a,

K-^a^ iK'+a,K-\-iK'-\-a^

here

a

is

infinitesimal,

and

A

denotes

-

a



16

MR.

GLAISHER,

TRANSFORMATION

AND

DEVELOPMENTS

Thus the column

headed

0

shows

that,

beinginfinitesimaly

sna

=

a,

n8a

=

-,

dsa

=

-,

c.

The column

headed

K

shows

that

en

(Jr+

a)

=

-

A?'a,

dc(^+a)

=

--,

c.,

and

80

on.

Digiti

zed

by



Goo

gle|'



18

PROF.

CATLBT,

SYSTEMS

OF

EQUATIONS

In

the

last column the values

of

the functions

for

xâ,

a

being

infinitesimal,

re

added.

By

the aid

of

the three

previous

olumns the values of

the

functions

2K- ay

2iir'+a,

2^4

2iK'

+

a

for

a

infinitesimal

may

be written

down

at

sight

by affixing

he

proper

sign

;

for

example

sn(2^+a)

=

-a,

sn(2iX'4a)=

+

a,

sn(2Z'+2iX'+a)=-a,

cn(2Jr+a)=:-l,

cn(2iJS:'

a)=-l,

cn(2JS:+2tJr'+a)=+lj

c.,

c.

SYSTEM

OF

EQUATIONS

FOR THREE

CIRCLES

WHICH

CUT

EACH

OTHER

AT

GIVEN

ANGLES.

By

Prof.

Cayley.

Consider

a

triangleBC^

angles.4,'B,

7

(-4

B-\-

Cîr)

:

to

fix

the

absolute

magnitude,

ssume

that

the

radius

of

the

circumscribed

circle is

=^1,

the

lengths

of

the

sides

are

therefore

=

2

sin^,

2

sin ,

2

sin (7

respectively.

n

the

three

sides

as

bases,

outside

of

each,

describe

isosceles

triangles

aBGy

bGAj cABy

the base

angles

whereof

are

=

a,

/8,

7

respectively.

f

we

draw

a

circle

touching

aJ5,

a

(7

at

the

points

-B,

G

respectively;

circle

touching

iC,

IA

at

the

points

(7,

respectively

and

a

circle

touching-4,

B

at

the

points

y

B

respectively

then

these

circles

form

a

curvilinear

triangle

BGy

the

angles

whereof

are -4+)8

+

7,

jB

+

7

+

a,

(7+a

+

)8

respectively.

aking

as

origin

the

centre

of the

circumscribed

circle,

nd

through

this

point,

or axis of

^,

an

arbitrary

ine,

ts

position

etermined

by

the

angle

dy

I

write

for convenience

F=0-]-2By

rê-Ây

^'

=

^

+

/S

+ 7,

?

=

^

+

2jB+2(7,

Gê^By

B^B+y

+

OLy

H=0y

E'=^0^2B+Cy

G'^C

+

a+fi;

then the

coordinates of the

angularpoints

Ay By

G

are

(cos-F,

injp'),cos

G,

sin

G^)

(cosS,

ini7)

respectively;

nd

the

equations

f

the

three

circles

are

(sin

-4

a)

A' /

sin

(-4

a)

.

-,A

sin'-4

x

+

\

^cos^')+{y

+

^^

ism^'j

=-t-^,

sma

/

V

sin a

/

sin a

'

/

8in(5-/S)

^Y

,

/

^sin(j5-/3)

^V

sin'5

[a

+

V--5

'cos ?

+

(y

+

\

^

^

sin

g

)

=-^^

\

sinjS

/

V sm/tf

/

sin'/S

/

sin(C-7)

TT'V

.

f

8in((7-7)

.

^V

sin'O

{x

+

-.

i^cosJ?

+

y

+

-'

^

sinJ?

=-:r-j-,

V

sin7

J

V

am

7

/ Bin'7

respectively.



FOR

THREE

CIRCLES.

19

In

verification

observe

that

we

have

0'H^2ir-2A,

?'-J?=-w+^,

(7-^'=27r-j4,

-F^A^

H^F^

-25,

W^F^

ir-^-B,

^-O'^

-5,

F^ ^B,

F^G^

-2(7,

^'-(?'=-,r+(7,

^H^

-C7,

Q^H^Oj

hence

(cos

7-

co8fi^)'

(sin

7-

sm ^)'-

2

-

2

cos((7-fi^,

=

2

(1

cos

2^),

=

4

sInM

;

and

we

thus

see

that

the

sides

are

2

sin^,

2

sin

5,

2

sin

0

respectively.

The

firstcircle

should

pass

through

he

points

cos

O^

sin

CF)j

(cosjB',

in

J?)

we

ought

tnerefore to

have

tor

the

first

of

these

points

^^^BinM- )

_

.inV- )^BJn^l

Sin

a

^

'

sin

a

sin

a

'

that

isy

.

.sin

(-4

-a)

.

.

sin (-4-a)

sinM

1+2

^cos-4

+

Vt

i

as

.

,

sin

a

sma

sin

a

Bnd for

the

second

of

the

points

he

same

equation.

Write

for

a

moment

^

vAuA

. sinf^-a)

^

-

X

=

:

,

then

:

'-

9

JC

cosa

*

cos^,

sin

a

'

sina

'

and the

equation

is

1

+2'(Xcosa-

cos-4)

cos^

+

(Xcosa-

cos-4)'=s-r,

that

is,

1

cosM =

-3l*

sin*a,

which

is

right.

The

second

and

third

circles

should

intersect

at

the

angle

A\

that

is

we

ought

to

have

\

smp

sin7

/

\

sin/i^

sin7

/

siu^JS

sin'C

^

sin

-B

sin

(7

.,

=

-^-fo

+

^ T

+

2

.

^

.

COS^

,

Sin

p

sin

7

sinp8in7

.

'

or

reducing

and

for

cos

(

O'

H')

substitutfng

ts

value,

=

cos^,

the

equation

s

sin'(5-/3)

.

8in'((7-7)

.

,sin(5-/3)sin((7~7)

.

jTB

*

5

T

^

i

7i

eos

^

sin

p

siu

7

sin

p

sin

7

sin*j5 sin'*

^

sin

sin

(7

=

-T-^TD

+

+2

.

^

.

cos-4.

sin

p

sin

7

sin

p

sin

7

C2



20

PROF.

CAYLET,

SYSTEMS OF

EQUATIONS

1

Writing

here

sm-B

^

sinJ?

'

^

Binp

'

siny

'

the

equatiop

s

(

Ycos^

cos

j5)*

(Zcos7

cos

Cy

+

2

(ycos/8-

co8j5)

(Zcos7-

cos

C)

cos-4

=

r'

+

Z'

+

2yZcos4',

Tiz.

this

is,

Y*

cos'^

+

Z*

cos*7

2

rZcos^ cos

7

cos^

-

2

Fcos^

(cos

5+

cos

Ccos.4)

2Zcos7

(cos

+

cos-4

cos

5)

+

cos^jB^-cos'C+

2

cos

-4

cos

5

cos

(7

4=r'

+

Z

+

2rZcosJ'.

Keducing

by

the

relation

-4

4

5+

(7=

ir,

this

becomes

2

FcosyS

sin-4

sin

(7

2^cos7

sin-4

sin-B+

1

cos-M

=

Y'

sin*i8

Z*

sln'7

2

rZ(cosud'

co3/3

cos

7

cos-4)*

Here

-4'

=

-4

+

^

+

7,

and thence

cos^' =

GOB

A

(cos^

cos

7

-

sin

^

sin

7)

Bin-4

(sin7

os^S

+

sin^S

C0S7),

and hence the

right

and

is

r'8in /3

+

Z'sin 7

2

yZ(cos-4

sin^

sin

7

+

sin

A

sin

7

cos 13

+

sin

A

sin^ cos

7)

or

reducing

by

FBin)8

=

sinJ5,sin7=sin(7,

this

is

=

sin'-iB

sin*

(7 2

sin B

sin

C

cos

A

2

y

cos)8

sln-^

sin

C

2Zcos7

sin-4

sinjB,

and the

terras

in

Yj

Z

are

equal

to

the

like

terms

on

the

left-hand

;

the whole

equation

hus

becomes

-

1

+

cosM

+

8in''jB+

in'

(7-

2

cos^

sin

j5

sin

(7

=

0,

where

the

last

term

Is

=

2

cos

-4

{cos

54-

(7)

cos

B

cos

(7},

=

2

cos*-4

-

2

cos

-4

cos

J5

cos

(7,

=

-

2

cosM

+

(cosM

+

cos'

+

cos'(7-

1),

;

=

cosM

+

cos*

J?

+

cos'

(7

1

;



PROF.

CAYLEY,

LEQENDRIAN

COEFFICIENTS.

21

the

equation

s

thus

-

1

+

cos*^

+

sin'J?+

sin'(7- co^'A

+

cos'5+

cos'C-

1

=0,

or

finally

t is

-1

+

1+1

1=0,

which

is

an

identity.

he

formulae

for the intersection

of

the

third

and

first

circles,

nd

for that

of the first

and

second

circles,

re

of

course

precisely

similar

to

the

above formula

for the intersection of

the

second

and

third

circles;

nd

the

verifications

are

thus

completed.

Cambridge,April

7,

1887.

NOTE

ON THE LEGENDRIAN

COEFFICIENTS

OF

THE

SECOND KIND.

By

Prof.

Cayley.

As

regards

he

integration

f the

equation

(l-*')g-2x|+ (

l)y-0

(n

a

positiventeger),

t

seems

to

me

that

sufficient

prominence

is

not

given

to

the

solution

where

P^

is the

Legendrian

integral

of

the

first

hind,

a

rational

and

integral

unction

of

x

of

the

degree

tz,

and

Z^

is

a

rational and

integral

unction of the

degree

n

1

;

viz.

we

have here

a

solution

containing

o

transcendental

funqtion

other

than

the

logarithm,

nd

which should

thus

be

adopted

as a

second

particular

ntegral

n

preference

o

the form

y

=

C

in

which

we

have

the

infinite

series

Q^

which is

an

unknown

transcendental function.

Moreover,

the

expression

sually

iven

for

Z^j

viz.,

^_2n-lp

,

2n-5

,

2n-9

*

l.n

- '* 3(n-l)

- ^5(n-2)

^'^

(to

term

in

P,

or

PJ,

is

a

very

simple

and

elegant

one;

but

the

more

natural

definition

(and

that

by

which

Z^

is

most

readily

alculated)

s

*c

+

1

Z^

isthe

integral

art

of

iP log

-,

when

the

logarithm



22

PBOP.

CAYLEY,

LEGENDBIAN

COEFFICIENTS.

is

expanded

in

descending

owers

of

x^

viz.

it

is

the

integral

part

of

W,h-\^--)

(whence

also

Q^

is

the

portion

ontaining

egative

owers

only

of

this

same

series).

The

expressions

for

P^,

Pj,...Pj

re

given

in

Ferrers*

Elementary

Treatise

on

Spherical

armonica^

cfec.

London

1877,

pp.

23-25.

Reproducing

these,

and

joining

o

them

the

values

of

^,

Z^...Z^^

e

have

as

follows

:

read

P,

=

fee'

i,

and

so

in

other

cases.

p.

=( ...

I)

\v

-

W

+

W

-

A

,

P,=( '...x)

vy.

_

^

+

a^ys

_

ij

,

P,

=(* ... )

Vi^

-

AJiit

+

ifji

-

Vi/'

+

P.

=(as'...a )i.Hgi-

Jfi

+

io^

-

Xj5i

+

IJI

,

II

,

V

+

A

,

z.

=( '...x)

vv

-

-4^

+

W

,

Z,

=(a/'...l)

^

-

iji

+

w

-

H

.

z.

=(x\..x)

v^

-

AV

+

V^ -^4Vy^

,

Z,

=(x'...l)ifi i-

-tflli

+

m^

-H\l^+

iff

.



MR.

LABMOB}

TBAKSFOBMATION

OF SUBFACB

INTEOBALS.

23

I

notice that

the

numerical values

of

P

Pj

...,

P,,

for

a;

=

0*00,

O'Ol,

...,

I'OO

are

given

[Report

of

the

British

Association

for

1879,

Report

on

Mathematical

Tables

)

;

as

the

functions

contain

only

powers

of

2

in

their

denominatorS|

the decimal values

terminate,

nd

the

complete

values

are

given.

The

functions

Z have

not

been

tabulated,

^

the

denominators

contain

other

prime

factors,

nd

the

decimal

values

would

not

terminate.

Cambridge,

Mareh

29,

1887.

THE

TRANSFORMATION

OF

MULTIPLE

SURFACE INTEGRALS INTO

MULTIPLE

LINE

INTEGRALS.

,

By

J.

Larmor.

An

integral

xtended

throughout

volume

can

in

various

ways

be

expressed

s a

surface

integral

ver

its

boundary.

Many

elegant

theorems

of

this

kind

have

been

givenby

Gauss.*

L

But

in

order

that

the

integral

ver

a surface,

f

a

vector

function,

eaning

thereby

the

integral

f

its

normal

component

over

the

surface,

ay

be

expressible

y

a

line

integral

ver

its

contour,

the

function

must

satisfy

certain

condition.

Li

fact the

integrals

ver

any

two

surfaces

abutting

n

the

same

contour would

then be

equal,

and

the

two

together

would

form

a

closed

surface,

uch

that the

integral

aken

in

the

same

sense over

the

whole

of it

would

be

equal

to

zero.

Now if

B

denote

the

vector,

X, Y^Zita

componentd

parallel

o

the

axes,

and

B

cose

its

normal

component,

//iieo... .///(f.f.f).

,)

Therefore

if

this

condition of

zero

integral

s

to

hold for

all

closed

surfaces,

e

must

have

identically,

hroughout

he

space

considered,

dX

.

dY

.

dZ

^

,-

iJ+^+5?= '

(')

as

the

condition

required.

*

Theoria

AUraetionit...,

omm,

Soe,

GUtvng.^

I,1813,

or

W^rke,

Band

V.



24

MB.

ULBMOBy

THE TRANSFORMATION OF

MULTIPLE

The truth of

the

formola

(1)

requires

hat

the

vector

should

not

become

discontinuous

or

its

differential

coefficients

infinite

anywhere

in the

space

in

question

for

if

that

were

not

provided

or,

he

integration

f its

ri^ht-hand

side

might

introduce other

terms:

cf.

Maxwell's

ElectricitVy

b.

I.

The

proposition

ust

therefore

be

applied

n its

simple

form^

only

when

the

region

in

question

oes

not

contain

places

where

the

vector

is

discontinuous

or

its

differential

coefficients

infinite.

If

Xy Yj

Z

are

the

components

of

a

flux

R^

the

condition

(2)

is

the well-known

Equation

f

Continuity,

hich

secures

that

the

flux is

that

of

an

incompressible

ubstance.

Thus

in

continuous motion

of

incompressible

luids

the

flux

through

any

ideal

aperture

is

expressible

s a

line

integral

ound its

contour

;

the

reason

for which is

obvious.

To

determine the

form

of the

integral

elation in

question,

we

may

firsttake

the

case

of

a

small

plane

surface.

Then

j{ada,

/3 7y)

^jjdxdy

g

|)

(3)

by

immediate

integration,

he rule of

signs

being

that

the

line

integralroceeds

ound

the

contour

in

the direction

from

^

to

y

in

the first

quadrant.

In

the

same

way,

for

areas

in

the

planes

of

yz

and

zx^

we

have

/0Si3,

, ).//* (|-f)

4)

/(r +.i.)-//.i (|.-|)

5)

By

what

precedes,

xpressions

o

be

integrated

n

the

right-hand

re

to

be

taken

as

the

components

normal

to

the

coordinate

planes

of the

vector

function

B]

and

we

remark

that

they

satisfy

2).

We

are

entitled

therefore

to

assert

for

any

small

plane

contour,

that

/(ada5^dy

+

7cfe)

^JJdS.BcoBB

(6)

where

the

components

of

B

are

^ dy^dz'

dz

dx'^^d^'Ty^' ^^



26

lOL

LAXXOB,

THB

TSAaSTOUULTIOV GP ITULTIFLS

We

nutjr

obtain

odmr

fonns

fiirAe theotem

hy

miiXmg

to

the

rigfat-hfliid

de

of

(9)

the line

int^nl

of

anj

exact

differential|

wbidi

will

add

nothmg

when taken loond

the

drcnit.

This

fomrala

(9)

expnaMs

as

a

line

int^^

the

flax

dae

to

a

single

sonroe

o

floid

at

the

orig:in

f

eoordinates,

r

the

induction doe

to

a

angleattractingarticle

tnated

there

;

and

from

it

any

more

general

case

might

be

deduced

by

summation.

Bat

dcTclopment

n this direction

amply

leads

to

the

well-Juiown

theory

of

the

Tector

potential

n

Electro-ynamics*

IL There

is

another

class

of

int^irsls

elated

to

Mathe-atical

Physics

n

which the

intends

are

extended

oyer

two

contoms.

For

instance,

uniformly

uminous

open

surface

emits

a

quantity

f

radiation

through

a

giren

aperture

which

depends

nly

on

the

contours

of

the

sur ce

and

apeiturei

are

being

taken

that

all

parts

of

one

contour

are

yisible

from

all

parts

of

the

other.

Again,

the

mutual

energy

of

two

closed

electric

currents

may

be

expressed

either

as

an

integ^

extended

over

their

circuits,

r as a

surface

integral

erived

from

the

equivalentagnetic

shells.

We

propose

now

to

investigate

he

general

forms of

such

relations.

If

a

line

integral

ound

a

contour

is

to

be

expressible

s a

surface

integral

ver

a

sheet

bounded

by

the

contour,

by

means

of

(6),

it

must

involve

the elements of the

contour

linearly.

Therefore

the

most

general

type

of

double

line

integral

n

question

ust

involve

both

contours

linearly.

he

function

to

be

integrated

an

only

involve the distance

between

two

elements of

the

contours

and

the mutual inclinations of

the

distance

and these elements.

If

r

denote

the distance

of

the elements

dsjds'y

nd

i^,

'

the

angles

t makes

with

these

elements,

and

e

the

angle

between

the directions

of

the

elements,

he

most

genersd

orms

therefore involve

only

//c c 7(r)cose,

(10)

and

JSdsda'

(r)

cos^ cos^'

(11)

Of

these

the latter

clearly

anishes when either circuit

is

complete.

The

former

where

V^

m',

vl

are

the

directioncosines

of da'

;

^Jds'ffdB{X\'\-

fi

+

Zv),by

(6),



SURFACE INTEGRALS

INTO

MULTIPLE

LINE

INTEGRALS.

27

where

\

fi^

v

are

the

direction

cosines

of

the

normal

to

dSj

and

r-/'(r)?^,

49,

y,

z

being

the

components

f

r,

the

origin

being

taken

temporarily

t

the

position

f

ds'.

Thus

changing

the order of

integration,

nd

transferrins

the

origin

o

the

position

f

dS^

so

that

we

write

^',

y

,

'

for

a;,

y, ,

we

have

jj,SJfir)

fj^a.'.^^^dy'

'^^^d.-)

^ffdsjjds'{X'\'

ry

+

v),

-;|:{;/'(r)|[-(y

O^V

+

x'X(y '

V)+...+

...]

-?/'(r)[XX'

/ /*'

' '']

^rj

|1/'

r)l

-

cos

*;

+

cos

^

cos

^

-

^/'

r)

os

*;

;|:{''/'(r)lcos,

r|^|i/'(r)|co. ?



28

MR.

LABMOR,

THE

TRANSFORMATION

OF

MULTIPLE

where

17

is

the

angle

between

the normals

to

dS^

dS

each

drawn

towards

the

positive

ide

of

the

surface,

nd

d^

ff

are

the

angles

between

these

normals

and

r,

whose

direction

is

the

same

in

both

cases.

Therefore,inally,

\\d8\\dS

r

1^/'

}cos ?cos^-l

{r/'(r)}

cosi;]

^jdsjdsj^r)

ose,..

.(12)

where

the

positive

ide of

the surface

is

determined

by

the

rule that

a

right-handed

crew

in that

direction

corresponds

to

the direction of the line

integral

ound it.

We

have

proved

that

this result is

the

most

general

possible

f its

class.

Particular

cases

may

be

noted

as

follows

:

(i)

Make

the

two

circuits

coincide.

(ii)

Make the

two

open

surfaces

coincide,

nd

we

express

the

double surface

integral

by

a

double

line

integral

ound

the

contour.

To

avoid

infinities,

'[r)

must

not

contain

powers

of

r

lower

than the

inverse

firet.

(iii)

ake

the

surfaces

plane,

o

that

17

is

constant.

(iv)

Make

/'W

=

-^;

then

jjdSJjdS'

^'^^^r^^

-

i

Jdsjds'

ogr

cose.

...(13)

The

left-hand

side

is the

expression

or

the illumination

from

S

that

is

intercepted

y

8'

when

the

brightness

f 8

is

unity;

and

it follows from

elementary

opticalrinciples

that

this

quantity

ust

be

expressible

s a

line

integral

round the

contours

of 8 and

8\

When S

and 8'

coincide,

e

have

27r5= -

^Jdsjdslogr

cose,

(14)

true

only

when

8 is

plane

;

for

when

8

is

not

plane

he real

optical

nterpretationails,

he

parts

of

the surface

not

being

in

full

view of

each

other.



8UBFACE IMTEOBALS

INTO

MULTIPLE

LINE

INTEOBALS.

29

(v)

Make

/(r)

=

-^,

o

that

/'(r)

-

J;

then

which

is

Neumann's

well-known

expression

or

the mutual

energy

of

two

simplemagneticshellsy

r

of

two

linear

electric

currents.

(vi)

Make

/'(r) Or;

then

/^5/(?/S'co8i7

-l/t /( Vco8e,

(16)

thus

giving

double

line

integral

orm for

JWdS^

where

n'

denotes the

area

of the

projection

f 8*

on

the

tangent

plane

at

dS, It

was

clear

h

priori

that

such

a

form

must

exist,

for

this

integral

epends

only

on

S

and

the

contours

of

8'^

while the

other

form

jUd8'

shows

that it

depends

only

on

the

contour

of

fi^;

hus

the form

of

the

function of

r

that

multi-lies

COSE

is

all that

remained

h

priori

o

be

determined,

nd

that

might

have been

found

from the

simplest

particular

ase.

When

one

surface

8

is

plane,

e

have

fi^n'

-i/i5/A'r'cose, (17)

where n' denotes

the

projection

f

/8'

on

the

plane

of 8.

Where

5,

8 coincide

in

one

plane,

e

have

8''

=

-ifdsJd8r^

OBB

(18)

And

comparing

this with

(iv)

e

deduce

(47r/8)

{Jdsjds

ogr

cose}

-

lir'

Jds

Jds

r*

cose...

(19)

for

any

plane

circuit

;

a

striking

esult.

The theorems

justgiven

may

be verified

by

direct

integra-ion

when the

surfaces

are

plane

circles,

and(18)

without

much

difficultly

or the

general

surface]

by applying

them

to

surfaces

bounded

by

other

curves,

we

obtain

evaluations

of

a

crop

of

definite

integrals

f somewhat

unusual form.

III.

If

elements of

three surfaces

enter

into

a

triple

integral,

he

components

of

the elements of their

three

contours

must

enter,

each

linearly,

nto the

corresponding

ine

integral.

The

most

general

form

of

such line

integral,

independent

f

special

oordinate

systems,

which

gives

finite

Talue

when

taken

over

completecircuits,

s

m{r,r\r )

dx

^

dy

J

dz

dx

,

dy'

,

e '

d7 \

dy'\

dz



30

MR.

JENKINS,

ON

THE

OBDER OF

PROOF

OF

THE

where

r,

r',

are

the

mutaal

diatances of the

three

elements

of

contour

;

and the

determinant is

equal

o

30

dsdi

dd\

where

*=sini(a

+

J+c)8ini(i+c-a)sini(c+a

J)

flinj(a+i-c),

a,

i,

c

ieing

the

sides

of

the

sphericalriangle

etermined

by

the

directions

of

rf ,

fo\

fo .

The

integral

ay

therefore

hj

application

f

the

method

of

II

be

expressed

s

a

symmetricalriple

urface

integral

the

general

formulae

are

long,

ut

the

degenerate

ases

would

probably

e

interesting.

Finally,

here does

not

seem

to

be

any

reason

why

the

considerations

on

which

these theorems

are

founded should

be

restricted

to

the three dimensions

a;,

y,

z

of

ordinary

pace

;

but the

more

general

results

would

probably

be of

only

analytical

nterest.

ON

THE OBDER OF

PROOF

OF

THE

PRINCIPAL

EQUATIONS

OF SPHERICAL

TRIGONOMETRY.

By

ilf.

Jenkins^

M.A.

The

principal

ormulas of

spherical

rigonometry

re

made

to

depend,

holly

or

partially,

n

three

independent

quations,

which

are

of

a

less

simple

haracter than

most

of those which

are

derived from

them

;

that

is

to

say

on

cosa

~

COS

cose

+

sin

h sine

cos^

with the

two

other

equations

f

the

same

form.

Independent

proofs

re

given

of the

more

simpleequations

sin-4

^

sin-B

_

sin G

^

sina

sinft

sine

^

but

as

these

constitute

only

two

independent

quations,

third

is

needed

for

the

complete

investigation

f

the

properties

f

a

spherical

riangle.

propose

to

take the

equation

sin

(A

+

B)

_

cosg+cosft

sin 0

1

+

cose

(which

in

the usual

order would be

obtained

by

the

multi-lication

of

the

expressions

f

two

of

Gauss's

theorems),

ive

an

mdependent

proof

of

it,

se

the

properties

f

colunar

and

of

polar

triangles

o

obtain

the

equations

f similar

form,

and

then

show

how

these

may

be

applied

o

prove

other

formulas.



PAINCIPAL

EQUATIONS

OF

SPHERICAL

TRIGONOMETRY.

31

Ab

a

leroma

take

the

equation

-C08a+C0R

cosm

2cos^c,

m

being

the median drawn from

G

to

the

mid-point

f AB,

Let

ABG

be

a

spherical

riangle,

the

raid-point

f the

arc

AB^

0

the

centre,

Ca,

00,

CS the

perpendiculars

n

OAj

OBj

OD

respectively,

/i,

v

perpendiculars

n

OA,

0B\

then

because

CaO,

G0O

and

CiO

are

rightangles,

he

sphere

on

OG

SLB

diameter

passes

through a,

^8,

S,

Therefore

O,

a,

ffy

S

are

on

a

circle;

nd

because

aOS

=

/905,

Sa^Sff,

also

Bfi^Sv]

hence

fia^v/B]

and

/a,

v

if

distinct

from

a

and

0

respectively

coinciding

hen

GA

GB)

must

be

on

opposite

ides

of

those

points

with

regard

to

0,

because

the

angles

OaS,

O0B

are

supplementary.

herefore

cosa

+

cos

J

OoL+ 00

20fi

^

,

=

j^

=

-7^=2cosic.

coswi

Oo

OS

*

We

may

note

that

the

same

proof

applies

f 2 be

not

the

mid-point

f

AB]

except

that,

instead

of

Sa

being

equal

to

S0J

we

have

a0

^

aS

^

0S

sin

c

sin

AD

sin

BD

'

also

OS.a0^

O0.a8+

Oa./3S,

whence

we

have

cos

OD

,

sin

c

=

cos

a

.

sin

^2)

+

cos

b

.

sin

BD.

Proceeding

o

the

equation

sin

(-44-

B)

cos

a +

cos

sin

C

1

+

cosc

'

let

D

be

the

mid-point

f

the

arc

AB^

join

OD]

producei

making

DK^

GDy

and

join

KA^

KB

so

as

to

obtain

the

spherical

homboid

A

GBK.

In

this,

s

in

plane

geometry,

opposite

ides

and

angles

re

equal,

and

alternate

angles

re

equal;

therefore the

angle

GAK=iA

+

B.

If

-4

4-5 7r,

CAK

is

a

proper

spherical

riangle

nd

denoting

D

by

m,

Bin

OAK

^

sin2m

__sin2m

Bin-4(7jr'

sin-^JT

sina

'

sin

A

CK

sin

^c

,

sin

A

sin

a

-

A~

-

-^-^

}

^^^

-^ n

=

^'

;

sm^

sm/7i

'

BiuU

smc



32

MR.

JENKINS,

ON

THB

ORDER OF

PROOF

OF

THE

whence,

by

multiplication,

Bin

{A-{-B)

2

cosm

2

cosm

cos^

Bin

(7

2

cos

Jc

~

2

cob'^c

__

cosa+

cos

J

1

+

cosc

If

A

+

B Wj

CAK is

not

a

proper

sphericalriangle;

but

since

CK is also

tt,

if

we

cut

off

from

it

CO'

=

tt,

and

treat

the

triangle

AG'

as

before,

and

note

that

sin

{A

+

B)

is of the

same

sign

as

sin

OJT,

we

see

that the

previous

quation

till

holds.

Next,

by

means

of

a

colunar

triangle

y

a

being

unaltered|

B

changed

into

ir

Bj c.,

we

obtain

sin

[A^B)

_

cos

J

-

cosa

^

sin

0

1

-

cose

'

from

the

polar

triangle,

hangingAîr-ây c.,

e

have

sin

(q

+

h)

_

ooÂ

+

cos-B

sine

1

-

cos

0

'

and from another

colunar

triangle,

sin

(a

I)

cos-B

-

cos^

,

sin^

+

sin B

sinG

\

'-

=

7=

,

also

-T

r-j-

=

-:

.

sine

1

+

cosU

'

BmaH-Bin6

sine

Hence

X

i/ i.r x

sin^+sinB sina+sini

.

^

sine

tani(^+5)

^j -^

=

inrr

sm

G.

cos^-FcosjB

sine

'8iu(a+6)(l

08(7)

C08i(a-i)

.,^

co8i(a

+

6)

* '

. ît/ i

.

px_(cos^-co8^)(sin^

+

sin^

which,

n

substitution

and

reduction,

ives

cos'^(q-ft)_,,

^

j-^

cos

*

O.

cos^c

*

Similarly

e

may

obtain

the

rest

of

Napier's

analogies

and Gauss's

theorems,

he

sign

in

taking

the

square

root

being

determined

by

the

consideration

that the

greater

side

is

opposite

he

greater

angle

in

a

colunar

triangle

s

well

as

in

the

original

riangle,

hat is

^

+

-7r of

the

same

sign

as

a

+

6

-

TT,

as

well

as

J

-

JB

of

the

same

sign

as

a

-

6,



CONTENTS.

PA

OB

On

the

transformation

and

developments

of

the

twelve

elliptic

nd the

four

Zeta

functions

(continued).

By

J.

W. L. Glaisher

-

-

-

17

System

of

equations

for three

circles which

cut

each

other

at

given

angles.

By

Prof.

Catlby

..-----

18

Not

on

the

Legendrian

coefficients

of

the second

kind.

By

Prof.

Catley -

21

The

transformation of

multiple

surface

integrals

nto

multiple

line

integrals.

By

J. Larmor

- -

-

-

--23

On the order

of

proof

of

the

principalequations

of

spherical

rigonometry.

By

M.

Jenkins

--------30

The

following

apers

liave

been

received

:

Major

Allan

Cunningham,

On

the

depression

of

differential

equations.

Prof. W.

Woolsey

Johnson,

On

the

second

solution of

the

differential

equation

of the

hypergeometric

series,

and

the

series

for

K\

',

Ac,

in

Elliptic

Functions.

Mr. F,

Morley,

On

plane

cubics

which

inflect

on

crossing

the

asymptotes.**

Articles

for

insertion will

be received

by

the

Editor,

or

by

Messrs.

W,

Metcalfe

and

Son,

Printing

Office,

Trinity

Street,Cambridge

NoTioR.

A

plate

will be

given

whenever

sufficient

diagrams

have been

received.



No.

CXCV.]

NEW

SERIES.

s^

iL

JUL

18

1887

[July,

1887.

THE

-MESSENGER

OP

MATHEMATICS.

EDITED

BT

J, W.

L.

GLAISHER,

So.D.,

F.R.S.,

FELLOW

OP

TRINITY

COLLEGE,

CAMBRIDGE.

VOL.

XVH.

NO.

3.

|

MAOMILLAN

AND

00.

1887.

r.

MBTOALPB

\

ASTD

BON,

/

Price-One

Shilling.



9

PBINCIPAL

EQUA^teKSLÔffieitKIAL

BIOONOMETBT.

33

To

obtain

cos

Cy

^

28in^co8(7 smB

sm(A+C)

+

sin(A-

C)

cos

(7=

T

;

:i

s=

r-: r

^:

5

-

2sm^

2

Bin

A

sin//

_

sinh fcosa

+

cose

cose

cosa|

cose

cosa

cosJ

2sina(

1

+

cosft

1

cos6

j

sinasini

For

right-angled

riangles,

f

C=i7r,

sin^

may

be

obtained

direct

from the

sine-aquation

co8c=

cosa cos

6,

by

producing

^Cto

A\

so

that GA'^

CA;

then

CB

being

the

median of

the

triangle

GA\

COSC+

cosc=2

cos(

cosa,

cos^

.

H

=*

cosa,

from

the

sine-equation

y

a

modificationof

figure

or

otherwise,

cos^

_

sin

((7

^)

_

cose

-f

cosa

_

cosa

cosft

+

cosa

sin5

sinjS

l

+

cos6

1

+

co86

=

cosa,

.

_sin((7+^)

_

sinJg

sin(C-f

^)

sin

0

sin

C

sinjB

__

sinft

cose

+ cosa

__

sinb

cosa

sine

l

+

cos6

sine

^

sin

ft

cos

c

_

tan

ft

^

sin

c

cosft

~

tan

c

'

sin^

sin

a

1

+

cosft

sin

a

tana

tanil

=

sin((7+^)

sinft

cosc+cosa

sinftcosa

sinft

'

Â

^n

sin(0+^)

sin((7-f-

B)

cosc

+ cosa

cose

+

cosft

cot-acotJ?=

\

p

^

1

:;

^

=

-

7

r

siujB

sm-4

1

+

cosft

1

+ cosa

=

cosa

cosft

cose.

The

equations

or

a

quadrantal

riangle

ay

be

found

in

a

similar

manner.

By

eliminating

ose

between

Bin(-4+(7)

sinft

cosa+cose

^siniA C)

sinft

cose-cosa

'

A

=

r-,

T-and

\ -r

=

-,

r-

sm^

sma

1

+

cosft

smâ

sma

1

cosft

we

should

obtain cosft

cos

(7s=

cot

a

sinft

cot^ sinG.

If

the

equation

containing

os-4

+

cos5

were

deduced

from that

containing

osa

+

cosft

analytically,

nstead

of

by

the

use

of

the

polar

triangle,

e

should have

to

determine

cos^,

cos

J?

separately

n

terms

of the

sides.

We

may,

however,

rove

the

former

equation

ndependently

hus

:

TOL.

XVII. 0



34

MR.

JENKINS,

ON

SPHERICAL

TRIGONOMETRY.

In the

figure

to

the

lemma

proved

above

the

angalar

equation

orresponding

o

the

lemma is

cos

B

sin

D

CA

cos

A

sin

BCD

=

cos

CD A

sin

C,

the

signs

being

most

conveniently

emembered

by

producing

BA

to

C\

and

measuring

the

angles

all

one

way,

that is

cosBsinjDC^

+

cos

CAC

sin

B CD

=

cos

CD

A

smBCA.

To

prove

this,

draw

the

arc

CN

perpendicular

o

ABy

CE

perpendicular

o

ON;

jEa,JBJ8,

i

perpendicular

o

the

planes OBC^

OCA^

OGD

respectively,

nd

EK

perpen-icular

to

OG. Then the

five

points

jB,

S,

a,

K^

P are

on

a

circle

with

EK

for

diameter,

n

a

plane

perpendicular

o

0K\

a^

:

/8S

:

aS

=

sinaE^

:

smlSES

:

sinoM

=

sin

a

:

sin

A CD

:

sin

J?CZ ,

also

Ea=^ECcosCEa

=

ECcosB;

Efi^ECcosCE/3=^

EC

co

Ay

-BS

=

EC

cos

CES

=

EC

cos

CD

A,

and

J5z./8S=^/3.aS

+

^.a/8,

whence

cos

BeinDCA

cos^

sin

J?CZ)

=

cos

CD A

sin

C,

Let ABC be

a

sphericalriangle.

n

BC

produced

ake

CE=CA]

join

EA

and draw

the

arc

FCD

bisecting

he

angleECA^

bisecting

A

at

right

angles

n

Fj

and

cutting

AB

produced

n D.

Then

8in(a b)

_

sin BE

_

smBAE

_

sinBAE

_

nin

DAF

^

sine

~

s'lnBA

sin

BE

A

sin CAE

sin

CAF

*

but

cos^J9Fsin

CAF--

cosAFCsin

CAD

=

cos

A

CF

sin

DAF,

and

cos

AFC

=0y

therefore

sin

(a

+

J)

_

cos

ADF

_

cos

ADF

sinA CB

sine

cosAGF

cos

A

CF

sin

A

CB

__

cos5sin^

CZ 4co8i48in5

CZ

_

(co85+cos^)cos^

7

cos

(^Tr

i

(7j

in

C

sin

i

C

sin C

^

cosB+cos^

1

COS

0

In

a

similar

manner

it

could

be

proved

that

sin

(a J)

_

cosB-

cos-4

sine

~

lH-cos(7

'.

April,

11,

1887.



(

35

)

ON

THE

SECOND

SOLUTION OF

THE

DIF-ERENTIAL

EQUATION

OF

THE HTPER-

GEOMETKIC

SERIES,

AND THE

SERIES FOR

K\

E\ c.,

IN

ELLIPTIC

FUNCTIONS.

By

Prof.

W.

Wbolsey

Johnton.

1. The

following

olution

of the

case

of

failure

of

one

of

the

ordinary

olutions

of

a

linear

differential

equation

of

the

second

order

in series

was

suggested

to

me

by

Mr.

Forsyth's

solution

of

the

corresponding

ase

in

Legendre'sequation,

Messenger

of Mathematics^

ol.

xvi.,

p.

162.

The

form

of

solution

is

interesting

s

giving

the

series

for

the

functions

K\

E

c.,

in

elliptic

unctions

at

once

in

the form

given

by

Mr.

Glaisher

{Gamh,

Phil.

Proc.j

vol.

v.,

p.

186),

and

accounting

or

the

factors

which

he

has

called the

adjuncts^

occurring

in

the coefficients

of

the

series

{Camb.

Phil.

Proc.^

vol.

v.,

p.

240),

2.

Denoting

x-^hy^^

e

suppose

the

differential

quation

*(^)y-a^ i(^)y=o

(1),

which

is

the

most

general

form

for which

a

relation exists

between

two

consecutive

coefficients of

the series.

The

equation

eing

of the

second

order

we

may

assume

and

i ^

(5)

=^

(5 c)

(^

-

rf),

where

^

is

a

positive

r

negative

onstant.

We

may

also

take

the

exponent

s

as

unity;

or

putting

=x'

and

z

-7-

=y,

we

have

sx ax

8

'

and

equation1)

becomes

which

is

of

the form

(^-a)(^-J)y-2^aj(5-c)(5-cZ;y

0

(2).

C2



36

PROF.

JOHNSON,

ON

THE

SECOND

SOLUTION

OP THE

We

shalltake

equatioD

2)

as

the

standard

form,

recollecting

that

when

8

is

not

unity

the

roots

a,

b

and

c,

d

are

to

be

divided

by

,

and x' substituted

for

x

in the

final

result.

3.

Putting

in this

equation

we

have

2{(m

+

r-a)(m

+

r-5)^^aj*^

-p{m

+

r'-c)(m

+

r-d)

-4^*^*}

0,

and

since the

coefficient

of

a; *^

must

vanish,

{m+r a) (m+r 5)^, p( n+r

c

1)(m+r-rf l)^^j=0.

This

gives

the

relation

between consecutive

coefficients,

A

_

(m

c

+ r

l)(m-e? +

r-l)

.

and,

when

r

=

0,

(m-a)(m-J).4,

=

0,

whence

w

=

a or

w

=

J.

Let

a

^ 6,

then

taking

ti

=

a,

and

the

solution is

^

a

ft

.

( -c)(a-d)

(o-c)(a-c+l)(ffl-rf)(a- ;+l)

-]

^

1.2(a-6+l)(a-6

+

2)

^^*''

^-J

Again,

interchanging

and

b,

the

second

solution

is

R

r,

(b~c-)(b-d)

(b-c)(b-c

+

l)(b-d)(b-d+l)

1

^

1 .2

(6

-o+l)(i-o+2)

^^*''

^-

J

To

conform

to

the

notation

of the

hypergeometric

eries

let

us

put

a

c=a

c=a \

o-J+l=7

J

(3),



38

PKOF.

JOHNSON,

ON

THE SECOND SOLUTION OF

THE

The

factor

a; **

a; .T*

a: (l

Alogaj+...),

nd

finally

he

remaining

actor

in

equation

^6)

is

a

function of

h which

we

shall

denote

by

^ (A),

nd

which is such that

by equation

4)

y,=ajXO).

The

complete

ntegral

6)

may

now

be

written

y

=

^^.+B,7;.,4~(l

+

Aloga:+...)a;''[t(0)At'(0)+...]

-î^,

+

5,7;_.+5y,loga;

Sx '(0)+ (8)

in which

A

is

put

for

the

constant

-4^,

-r*

and

the omitted

terms

are

terms

which vanish with

h.

It

remains

to

express

^'

(0)

as

a

series

in

powers

of

x.

5.

Let

H^

denote

the

coefficient of

{pxy

in the series

-^

(h)

;

that

is,

let

i7 i .

^_(a+A)...(a

r-l+A)(/3-h^)...(/3+r-l-hA)

/i-i,anax2-

(i

+

A)...(r*)(7

+

A)...(7

r-

1 +

A)

Then

f

(A)=S.

S:

(pa;)',

nd

^'(A)=2^

^'

(pa;)'.

ow

_p.r

1

1 1 1 1

1

^^^L +A

â+l+A'

+r-l+A

*

/3+A'^* '*'/3h-

-l+Aj

When

A

=

0,

this

becomes

^sîrj_

.

j^

1

L.1

La

+

s ^/3

s

1

+

5

7

+

5J

in

which

H^

now

denotes the

coefficient

of

(^pxY

in

the

series

FQi^

ffj

,

j?a;),

quation4).

Thus



EQUA.TION

OP THE

HTPERaEOMETRIC

SERIES,

C. 39

and,

writing

the

completeintegral

n

equation

(8)

in the

forms

y

=

Ay^

+

JBy,,

e

have for

the

two

independent

nte-rals

when

7

is

an

integral,

and

,. ly

(7-2)1

(y-1)

y

,

,v

where

^-' Ll-7U

/i

1

7/^^

1.2.7(7+1)

/I

.

1

.

1

.

1

111

-

1

\

/

M

,

1

(10),

and

Ty_^

denotes

the

sum

of the first

7

1

terms

of

y,

in

equation(5).

When

7

=

1

we

have

of

course

31

=

0,

and

when

7

=

2,

r,=a;-^^^

a;-\

Thus the second solution consists of three

parts,

the first

of

which is

the

product

of

the

firstsolution

by

log

a?,

the

second is

a

finiteseries

beginning

ith

o?* ^**

nd

ending

with

the

power

a? *,

nd

the third

is

the

secondary

series

y',

which

isthe

same

as

y^

except

that each

coefficient

is

multiplied

y

a

factor which

we

shall callits

adjunct^

onsisting

f

the

sum

of the

reciprocals

f the factors in the

numerator

taken

positively

rid of those in the

denominator

taken

negatively.

The firstcoefficient

of

y^

which

is

1

must

be

considered

as

having

the

adjunct

ero.

6.

The law

of

the

adjuncts

ust

stated is

the

same

as

that

pointed

ut

by

Mr.

Glaisher

in

the

case

of the

series for

JT',

E^

c.,

in the

paper

cited in

1,

although

the

notation

differs

from that of

the

hypergeometric

eries.

The

reason

for

this

persistence

f

the

law

is

readily

explained

s

follows,

and

is

illustrated in

the

examplesgiven

in the

succeeding

sections.

In

the first

place,

f

a

and

^8are

fractions

having

the

same

denominator

tw,

say

a

=

,

/8

,

the

coefficients

n

y^,

m m

m m

m\m

)

m

\m

J

^TT '

*

1.2.7(7+1)



40

PROF.

JOHNSON,

ON THE SECOND

SOLUTION

OF

THE

may

be

written

in

the

form

m.rny

'

9n.27n.7117

ray

+

tn)

If

the

adjancts

e

now

formed

by

the

same

law

as

before,

each

term

of

each

will

have

one

Tnth

of

its former

value,

and

the result

will

be

the

value

of

u\

so

that

the law

holds

good

for

the

integral

y^j

when the

coefficients

re

written

in

the

new

form.

Again,

if

we

have

occasion

to

introduce

new

factors

into

the

numerators

or

denominators of the coefficientsin

y^^

so

that,

for

instance,

e

write

j^y\

or

the first

solution,

hen each

adjunct

fformed

by

the

same

law

should

contain

the

additional

terms

7

.

If

then

we

take

t

y,

+

(

t)

i

for

the

second

integral,

he

law

of the

adjunct

ill still

hold

good.

7. We

proceed

o

apply

the formulae

given

above in the

case

of

the differential

equation

atisfied

by

K

and

K\

the

independent

ariable

being

the

modulus

k.

This

equation

s

*(l-*0g

(l-3*')|-%

O.

Multiplying

y

k

and

putting

-^=.9,

hence

this

becomes

yy-A; (y

+

2^+l)y

=

0;

and

putting

5=i*

to

reduce

it

to

the

standard

form,

equation

(2),

e

have

Here^^l,

and

a

=

0,

a

=

0,

6

=

0,

a

=

i,

whence

,

'

?=-i.

7

=

1-

:



=yi=i

+

^.* +^ **+-,

EQUATION

OF THE

HTP RaEOMETRIC

SERIES,

0.

41

Equations(4)

and

(9)give

then for

our

two

integrals

l^.=yilogi +

^*(2+2-

-

1)

*

+

^^]l;P^(2+2

+

|-l-l-4-i)**+....

It is

cnstomaiy

to

write

the

first

series,

hieh

is

the

alue of

,

in

the

form

IT

and

accordingly

e

write the

value

of

^y,,

which is

in

which

the

law

of

the

adjuncts

is followed.

Now

jr'=y.

log4

iy,

so

that the

law

holds for

the

series

for

K'

which

Mr.

Glaisher

writes

in the

form

^'=

log|

j

(log~Hl)*'+^(logf-f+

8.

Consider

next

the

equation

atisfied

by

E and

/',

hieh is

*(i-A^) +(i- o|+*y=o,

or

3'y-k\y-l)y

= 0.

Fatting

=

A^,

this

becomes

where

again

^=

1,

and

a

=

0,

a

=

0

6

=

0,

a

=

-i,

whence

d

=

-i,

7=1.



42

PROF.

JOHNSON,

ON THE

SECOND

SOLUTION

07 THE

Thus

y,=i+:itiA.+=iifc|iA*+

and

+

^^[^(-2+2+

+

1-1

-l-i-i)A*+....

Bat

y,

which

is

the

value

of

is

usually

ritten

2E_

_

1

r.3

1'.3'.5

,j,

in

which

we

have

dropped

from

the

numerators

of the

coeffi-ients

the

factor

1

which

corresponds

o

the

firstfactor

(

i)

in

the

coefficients

as

firstwritten.

Accordingly

f

we

write

-^(-i

i

+

i

+

J-i-i-i-i)*'

,

the

law

of

the

adjunct

s

not

followed

because

the

term

-

1

in

each

adjunct

is

now

superfluous;

ut

adding

y^

we

get

rid

of these

terms

and

have

the

integral

y.+iy,=i+iy.iogi'-^ (i-i-4)A'

-^(i

+

i+i-i-i-i-i)**-...,

in

which

the

law holds

good.

Now

/'

is

found

to

be

y^

log4

-(^^4-

iy,),

nd thus satisfies

the

law of the

adjuncts.

9. The

equation

atisfied

by

I

and

E^

is,

hen

aj

=

^,

in

which

a=l,

o

=

l,

6

=

0,

a

=

i,

,

whence

^

.

^=-i,

7

=

2.



44

PROF.

JOHNSON,

ON

THE

SECOND

SOLUTION

OP THE

in which

a

=

2,

a

=

2,

i

=

0,

a=f,

whence

^

.

rf

=

J,

7=3.

Thus

y.=**[i+a*'+rlli

*+ ]

and

y.=y.log* --^^^

l+^fe*-^*

+i*[f|(*+*-^-* *'+iill4(

the

expression

or

Ty-i

in

equation9)

in

this

case

containing

two terms.

The

value of

-

^

^^'^

^

is

iy

thus

^

-iy.-2.4*+2'.4.6*+-'

and

if

we

write

accordingly

he

value

of

i^y,

we

have

i^ry.

iSy.

log

A'-2(l-ii')

+

ig(i

i-i-i)

* +...

In

which,

in

order

to

follow

the

law,

the

terms

1

+

I

-

i

-

i

are

needed

in

each

adjunct.

Adding

therefore

f

of

the

preceding

eries

we

have

iVy,+

/jyi=iVyt

log**-

2

+iA'

in

which

the law of

the

adjuncts

olds

for

the coefficient

of

k^

and

higher

owers,

but

not

for the coefficient

of A;'.

But

this

term,

as

we

have

seen

above,

is

part

of the

expression

or

T^

and

not

of

that for

y'.

The

value

of

JT

+

G^'

is



46

PROF.

JOHNSON,

ON

THE

SECOND

SOLUTION

OF

THE

where

T^

denotes

the

sum

of

the

first

n

terms

of

the

preceding

series for

y

and

^-' [TFn-)('*^)* '

i.2(n

i)(/i

2)

(^

*

*

^^TTi

*

^TTa)

^ '^^

^ J

in

which

the

terms

in

the

adjuncts

corresponding

o

the

a-

and

/3-factor8

anish

because

these

factors

are

infinite. This

solution

agrees

with

that

of

Hankel,

referred

to

bj

Mr.

Forsyth

in

the

paper

cited

in

I.

12.

When

each

of

the

functions

4

nd

4 ^

is

of

the

second

degree

we

can

derive

a

series

either in

ascending

or

in

descending

owers

of

x.

For

example,

Legendre'squation,

(l-a,')g-2xg

n(n-H)y

=

0,

may

be written

in

which

p

=

1,

and

=

2

;

thus

a

=

i,

a

=

i,

*

=

^'

.

.hence

-=*(^- '

rf=-i(n+l),

7

=

1,

and

bj

equations

4)

and

(5)

the

integrals

re

,

r,

,

(1

- )(2

+

)

.

(l- )(3- )(2+ )(4+ )

-j

J^'-'L^ ^

2:3

^

2XI5

-e+'-j,

and

_

1

,

-n(\+n)

-n(2- )(l+ )(3

4- )

13.

But

if

we

write

the

equation

n the

form

(5-n)(5

+

n

+

l)y-i

(^-1)^

=

0,



EQUATION

OF

THE

HTPERGEOHETBIC

SEBIES,

C.

47

we

may

take

=

-

2,

and therefore

J

=

-in,

a=^(n

+

l),

whence

^

,

,

c=0, /8

=

Kn

+

2),

and

the

integrals

re

y -^

L

2(2w

+

3)

^

(w

+

l)(r 2)(n

+

3)(n

+

4)

I

*

2.4(2n

+

3)(2n

+

5)

'^

'

y

and

_

fi

w

(n

-

1)

^

w(w-l)(n-2)(n-3)

^

1

^''-^

L^ 2(2n-1)'^

^

2.4(2n-l)(2n-3)

^-J

14. When

n

=

i,

7

=

1,

and

the

two

series

are

identical

;

and when

2n

is

a

positive

dd

integer

is

an

integer

reater

than

1,

so

that

the

series

y^

contains

infinite

coefficients.

In

these

cases,

therefore,

e

have

by equation

9)

or,

taking

\y^

as

the

integral,ecause,

s

explained

n

6,

\y'

will

follow the

adjunct

law

when the coefficients

are

written

as

in

the

expression

or

y^

above,

sy.-yiiogj,

{ ( -2)(n-4)...(l- )}'

^o*k^iy^

where

T^^^

denotes the

sum

of

the first

+

i

terms

of the

preceding

eries for

y,,

and

*^

L

2(2n

+

3)

U

+

1

+2

'

2n

+

3/

- )(

+

2)(n

+

3)

(n

2.4

(2 i

3)

(2n

+

5)

(

+

l)(

+

2)(n+3)(n

+

4)^^^^ ^

J



48

PROP.

JOHNSON,

ON

THE

SECOND

SOLUTION

OP

THE

where

the

symbol(ad)

written

after

a

coeflSci^nt

enotes

its

adjunct.

This solution is the

same

as

that

given

by

Mr.

Forsyth

in

the

paper

before cited.

When

2/2

is

a

negative

odd

integer

he

solution

may

be

found

in

like

manner,

or

at

once

by putting

n

1

in

place

of

n

in the

solution

just

found,

he result

being

^

-^-

n

,

-n(l-n)

-n(l- )(2-n)(3-n)

1

Vi-'^

L^+2(l-2 )*

2.4(l-2n)(3-2n)

*

+-J

and

iv

-V

Wl

-

{(-2 -3)(-2 -5)...4.2}'(-2 -l)

in which

T_^^

denotes

^.

r

(n

+

l)(

+

2)

*

L

2(2

+

3)

*

+

+

(

+

l)(

2)...(- -3)

1

^2.4...(-2n-3)(2n

3)...(-4)(-2)

J'

and

*^

L2(l-2n)^'^^^

2.4

(1-2/1)

3-2/1)

^^^^

+-J

15.

The

functions

^

and

4 ^

in

equation

1)beingsupposed

real,

he

case we

are

considering,

amely

that

in

which

7

is

an

integer,

annot

arise

when

a

and

,

the

roots

of

^,

are

imaginary

but

c

and

J,

the

roots

of

ff ^^

ay

be

imaginary,

and

a

and

/3

will

then

take the

forms

a

=

/i

+

tV,

jS^fA--

tV.

In

this

case

equations4)

and

(5)

may

be

written

in

the

forms

yi

-A

1

+

/*'+ '_,,(m'

+

i^'XC/^O'+v'}

1.7

| J

+

1.2.7(7+1)

(px)'+...

W,

(^+1-7)'

+

^'

'

(2-7).

I

^

.

=

x--[l

+

(2-

7)

(3

-7)1.2

(i'^J+'-J-CS),



and

for

the

case

in

wtucb

y

is

an

integerquation

9)

becomes

(

SEP

26

1887

)

EQUiLTXOM

OF

THl ^ K ^|[^ ^UHt5sEBIE8,

C.

49

for

the

nes

.f

ty

(7-l)l(7-2)t

y

,

,0^

^

^

i*'-'

(/*

-7)'+

''l-.K/i-O'+F')}

'^^

' ^

^'

where

*

1.2.7(7+1)

V^

v

(m+O'

+

v'

*

-^-7Ti)(i'-)'+-]

These

forms,

ith

a

change

f

sign

of

v',

are

also

con-

venient

when

a

and

/9

are

irrational

real

numbers.

16.

The

form

of the

solution

when

7

is

an

integer

undergoes

odification

when

either

a

or

/3

is

an

integer

less

than

7.

In

the

first

place,

uppose

one

of

them

to

be

a

positive

nteger

ess

than

7.

We

cannot

now

employ

equation

9),

ecause

the

coeflScient

of

Ty-i

is

infinite;

n

fact,

in

equation

7)

is

now

zero,

and

the

complete

ntegral

(8)

reduces

to

m

which

7

is

a

finite

series

containing

a

terms.

It

is

to

be

noticed

that,

omparing

his

with

the

general

ntegral,

T

is

not

equivalent

o

y,,

but

we

have

in

which

the

coeflScient

f

y,

takes

the

indeterminate

form,

but

has

a

determinate

value

when

7

and

a

are

functions

of

a

single

uantity

nd

become

integers

imultaneously.

This

case,

in

which

the

finite

seriesis

not

the

limiting

value

of the

infinite

series

from

which

it

is

derived,

ccurs

in

the

solution

of

Kiccati's

equation

see

the

Memoir

by

J.

W.

L.

Glaisher,

hil.

Trans.^

881,

p.

771).

YOL. XVII.

E



50

PBOF.

JOHNSON,

ON

THE

SECOND

SOLUTION

C.

17.

In

the

next

place,

et

a

or

)8

be

zero or

a

negative

integer,

ay

a

=

n

;

then

y^

is

a

finite

series

of

which the

last

term

is

.

-^fi(-n+l)..,(^l)/3(/3-H) .(/3-hfi-l)

fil7(7

+

l)...(7+ -l)

^^^

In

this

case,

supposing

an

integer,

he

series

y

in

the

second

integral

s

not a

finite

series,

although

he

series

y^

from

which it

is

derived

is

finite.

Denoting

the

term

of

y^

just

written

by N{pxf^

the

coefficientis

.

.

/8(y8

l)...(g+ -l)

^^ ^

7(7+l)...(7

n-l)'

or,

when

fi

=

0,

-^^=1.

The

coefficient of the

next

term,

that

is

to

say

of

the

first

term

in

y^

which

vanishes,

s

the

correspondingdjunct

in

y

contains

the

term

,

which

is

infinite,

he

remaining

erms

being

finite.

Thus

the

entire coefficient

of

d^a-) *^

n

y'

is

(n+l)(7

+

7i)*

In

like

manner,

every

succeeding

djunct

ontains the

same

infinite

term,

and

the

entire

coefficient

in

y

is

the

same

that

it would be in y, with the omission of the

zero

factor.

Thus,

in

addition

to

the

part

of

y

formed

by

means

of

adjuncts

it m

the

actual

terms

in

y^,

we

have,

corresponding

o

the

vanishing

part

of

y^,

the infiniteseries

(

+

l)(7

+

)

L

(

+

2)(7

+ n

+

l)^

1.2.(/3 +l)(j8

+ w

+ 2)

(n+2)(n

+

3)(7

+

n+

l)(7+n+2)

ipxy+.



52

MR.

F.

MOELEY,

ON

PLANE CUBICS

WHICH

INFLECT

With

the first

value

for

m

the cubic is

y

(/

+

^\^y

+

3a,^ )

36,

(y*

3a, ')

0

.

.

.

(3).

Equation

(1)

now

becomes

-

8^(y

+

36,:ry 3a/y)

-

3

(-

\ay

+

{â^fiâ,')'y

+

2 Aa3^y^-.aAy)

0,

or

\

{\xy'

+

a,^'y)

hfl^x''

(2a, 3

a.)

x*y

+

2

Jôr/

=

0,

or

3

JgOry'

Sajb^x^^

or

y'

=

V'

(4).

Since

for

3

real

inflexions

a,

must

be

positive,

rite

a^

=

a*.

Then

the

lines

to

the other

inflexions

are

^

=

axy

so

that

they

are

harmonic

to

the

axes.

Substituting

rom

(4)

in

(3),

or

4y

+

3J,ar+12J,

0

(5),

which

is

the

line

of

inflexions.

Let

PQR

be

the

inflexions,

being

on

the

asymptote,

and

let the

inflexional

tangents

form

a

triangle

qr.

Let

0

be

the node and

let

FQB

and

Fqr

meet

the

ar-axis

in

T

and

L

?

.,^

It

is

calculated

at

once

that the

tangent

at

P is

3 -ar

+

y

+

3J3

0

(6),

and

at

Q

and R

3J,x-8(y+3J3)

9^y.

.(7).

Hence

Itis

manifest that

p

is

on

the

x-axis

;

also

putting

y

=

0in(5),(6),(7),

Op= 2.0T=8.0t.



ON

CROSSING

THEIR

ASYMPTOTES.

53

Also

from

(6),(7),

Or

and

Oj

are

y

Zax

......(8),

eo

that

if

OQ^

OR^ Oq^

Or

meet

the

asymptote

in

OQ,^OIt,^30q,

=

30r^.

The

equation

to

pQ^

is

1

=

?* ^

or

3J,

3J,x~8(y

+

3J^-^-,

(Box

-y)

=

2

(Sb^x

4y

+

12 J

J.

Hence

pQ^^

Or

and

similarly

?5

Og'

meet

on

P^J?.

These

are

of

course

not

all

indepenaent

esults.

2.

If

the

cubic

inflects

on

crossing

second

asymptote,

say

at

0,

then

Oq

is

parallel

o

the

asymptote,

o

that

y

+

Sax

is

a

factor

of

y'

+

3i,^y

+

Sa'x*.

Hence

the other

factor

is

j/

+

ax^

and

3 ,

4a.

Hence

y

+

ox

=

0

meets

the

curve

at

infinity^

r

H is

at

infinity.

The

metrical

relations

are

now

very

simple.

Let

OAy

OB

be

parallel

o

the

asymptotes

at

P,

Q.

Let

PFaqj

QFffp

be the

mflexional

tangents,

then,

mce

0p

=

20A=^S0a

and

Oq^20B=SO^f



54

MR. F.

MOBLEY,

ON

PLANE

CUBXCS

WHICH

INFLECT

aP^pq^

AB

are

parallel,

nd

Pi

2-45=

34P=

35$,

therefore

BP=PQ=:

QA

;

hence

pB^

OQ

are

parallel,

nd also

qAy

OP.

Also

in

the

asymptotes

meet

in

E^

PDxDA^PEx

OA^PQxAB^l

:

3,

therefore

PD

=

\PQ

=

QD.

Also

PQ

is

cut

by

OD

in

a

ratio

=

op :

eJS?=a^

:

PQ^^F:

FQ,

therefore

-Flies

on

OE.

And

since

EDiDO=ÊQ\BOÛZ

and

0F\

FEÔp:

QE^Z\4.

it

follows

that

DF\

FO^

OF:

FE

or

OF'^DF.EF.

.Clearly

ll

the

lines

in

the

figure,

nd

all

got

by

joining

the

points

n

the

figure,

re

cut

in

very

simple

atios.

(jriven

0,

P, Q,

the

construction

for

the

asymptotes

is

:

take

on

PQ

points

^

B such

that

BP=PQ^

QA^

then

OA^

OB

are

in

the

direction

of

the

asymptotes.

For

the

tangents

:

draw

Bp

parallel

o

OQ

to

meet

AO.

Then

pQ

is

the

tangent

at

Q.

The

cubic

cannot

inflect

on

crossing

ll

asymptotes

unless

each

inflexion

is

at

infinity,

nd

the

asymptotes

are

then

the

inflexional

tangents.

3.

A

non-singular

ubic

has

of

course

much

more

freedom.

But

if

its

inflects

on

crossing

ach

asymptote

the

line

of

inflexions

has

an

envelope

hich

it

is

proposed

o

investigate.

Let

J5r=ar+y+ =0

be

the

line

at

infinity,

=ax+by-\-cz=0

be

the

line

joining

he

points

here

the

asymptotes

meet

the

curve,

and

let

xyz

=

0

be the

asymptotes.

Then the

cubic

is

LKÂ-^mxyzÔ

(9).

The

first

differential

oeiBcients

are

aJ5r*+

22iir+

6my ,

...

.

The

second

are,

omitting

factor

2,

2aX'+i,

...,

(6

+

c)

/iL-t-i

+

3mx,

....



ON

CROSSING

THEIR ASYMPTOTES.

55

The

Hessian formed

.from

these

lastsix

works

down

to

2^{(a-J)(a-c)x+...}

+

6mK{(h

+

c)yz''

ar'

-f

...}

+

37ni(2y -ar'+...)

+

18m'jry

0

(10).

Letar

=

0

and i

=

0,

therefore

2

(y

+

)'

{(ft-c)

ft-a)y

+

(c-a)

(c-

J)z]

+

6in(y

){(

c)y5?-

y'-c *}

0,

or

writing,

ince

iy

-f

c

=

0,

y

=

c,

=

6,

(J-c)'{-c(J-a)-J(c-a)}3m(6-c){-Jc(J+c)-Jc'-cJ'},

or

(b

c)'

{2 C

a

(6

+

c)}

GrnJc

(J

+

c)

0

...

(11),

i

cs=0

would

mean

an

inflexionat

infinity].

Suppose

two

such

conditionshold

:

(a-c)

{2ca-

J

(c+

a)}

+

6mca

(c

+

a)=:

0,

(a

-

by{2ba

-c(b +

a)}+

Qmab

(ai'b)=^

.

Subtracting

nd

dividingy

c

-

J,

ic(2a-J-c)

+

6ma(6

+

c)+6wa'

+

3a'~2a (2c

2J)

+

2(6VJc

+

o')a-aJc=:0.

Let

a,

i,

c

be

the

roots

of

x*-px^

+

qx-r=iO

(12).

Then

since

3aic+6ma(a+J+c)+3a'-4a (

3)f

0(^+0*)-

Jc( +c)=0,

therefore

3r

+

6map

+

3a*

4a'

(p

-

a)

+

2a(2 *-2j-a )-^(; -a

0,

and

if the

3

such

conditions

hold,

,

,

e

are

given

by

4rar

+

Gmp^'

+

5ar*

4;?x'

2ar*

(p

2

j)

pr

=

0.

Dividing

his

by (12)

the

remainder

is

3

(/

-

3j

+

2mp)

x'

+

(yr

-;?j)

r,



66

ME. F.

MOBLET,

ON PLANE

CUBICS

WHic^

INFLECT.

and

therefore

the

only

conditions

which

the

constaats

must

satisfy

re

37

'

Jc

+ ca

+

ai-a'-

c'

2m

=

-*

^

s=

7

,

p

a-\-o-\-c

'

and

pq

=

dr

(13),

or

a(J-c)'

+

(c-a)*

+ c(a- )*=b.

Given

the

asymptotes,

he

envelope

f the Hne of inflexions

is

the

envelope

f

aar

-f

iy

-h

C2

=

0

subject

o

(13).

It

may

of

course

be

got

by

eliminating

,

but the

followingymmetrical

method

gives

the

result

in

a

better

form.

Equating

dif-erential

coefficients

with

regard

to

a,

J,

c,

and

neglecting

a

constant,

;r

s=

2

+

(J

+

c)p

-

9bc

or

pa*

(2?'

2

-

^)

a

+

9r

=

0,

and

a'-2?a'

ga-r

=

0.

From

these

we

must

eliminate

a.

Multiply

he

second

by

9

and

add

;

9a*-8pa-(p'-8g'-a:)

0,

therefore

{9

(p

+

2

-

a:)

8;?'}

(p*

j

-

x)

(^

-

Sg

-

x)

+

72pr}

{p

(

p*

82

-

ar) 81r}',

or-9a:^+18a:'(/-32)-^^(V+lS/?-9.152*

8p*2-722'=0,

or

x^'-Fx^+Qx^B^O,

where

i'=2(p*-32),

Q

=/

+

2/2

-

152' (/

-

32)

(; 52),

5

=

1

(/2-9A

therefore

y'-32

=

iP,



MR. S.

ROBERTS,

NOTE

ON

CERTAIN

THEOREMS.

57

therefore

( -*^)^+ )

^4'

or

(4g-P )(12 ?

+

P0

=

288PH

(14).

The

envelope

s

therefore

a

quartic.

It

meetg

the

line

at

infinity,

=0,

on

^

=

0,

which

is

the

minimum

ellipse

bout

the

triangle.

It

meets

the

sides,

=

0,

on

4^

=

P*,

but

this touches

at

the

middle

points,

nd

also

on

12

Q

-i-

JF*

0.

Hence

it

touches

a:=0

at

the

middle

point

nd

cuts

its

where

12y

+

(y+i5)

0,

r

y

+

70

=

4^30.

Hence

the

shape

is

as

in

the

figure.

Bath

College.

NOTE

ON

CERTAIN

THEOREMS

RELATING

TO

THE

POLAR

CIRCLE

OF

A

TRIANGLE

AND

FEUERBACH'S

THEOREM

ON

THE

NINE-POINT

CIRCLE.

By

Samuel

BoberU.

1.

If with

reference

to

a

triangle

he

centres

of the

cir-umscribed

and

inscribed

circles

and

the

orthocentre

are

respectively

,

7,

P,

and

the

radii

of those circles

and

the

polar

ircle

are

respectively

B,

,

p,

then

the

following

ell-

known

relations

hold

:

0P;

=

i?'+2p ,

7P =.2r'

+

p'.



60

PROF.

CAYLEY,

ON

THE

SYSTEM

OF

THREE

CIRCLES

It

is

easy

to

establish

also

without

direct

reference

to

conies

the

following

heorem

:

If

A^

By

C,

D,

E^

F

denote

six

points

or

lines,

nd

that

any

four

of the

points

onnect

equi-anharmonically

ith the

remaining

two,

or

any

four of the lines intersect

equi-

anharmonically

ith the

remaining

two,

and

if

any

five

of

the

points

ie

on,

or

any

five

of

the

lines touch

a

circle,

he

remaining

point

lies

on,

or

the

remaining

ine

touches

the

same

circle.

Now

let

ABG

be

a

triangle

elf-conjugate

ith

regard

to

a

circle

(/S).

Take

any

point

D

on

the

circle

circum-cribing

the

triangle

BG

and draw

the

polar

of

D

with

respect

to

(/S).Suppose

that

one

of

the

intersections

of

this

polar

ith

the

circle

circumscribing

he

triangle

s

E,

Then

take

F

the

pole

of

the line

joining

,

E

with

respect

to

(/8).

The

triangle

EF

will

be

self-conjugate

ith

regard

to

(5),

and

by

the

above

given

theorems

F

lies

on

the

circle

through

ABGDE^

and

must

be

the

second

intersection

of the

polar

of

D

with

that

circle.

This

is

theorem

(I),

nd

theorem

(II)

can

be

similarly

roved.*

ON

THE

SYSTEM

OF

THREE

CIRCLES

WHICH

CUT

EACH

OTHER

AT

GIVEN

ANGLES

AND

HAVE

THEIR

CENTRES

IN

A

LINE.

By

Prof.

Cayley,

In

the

system

considered

in

the

paper

System

of

equations

for three

given

circleswhich

cut

each other

at

given

angles,

Messenger

t.

XVII.

pp.

18-21.,

we

may

consider

the

particular

case

where

the

centres

of the

circles

are

in

a

line.

The

condition

in

order

that

this

may

be

so

is

obviously

sin

{A

a)

cosF\

sin

[A

a)

sini^',

in

a

=

0,

sin

{B

-

/3)

os

0\

sin

{B

-

/3)

sin

G\

sin

^

sin

((7-7)

cosjH',

in

((7

-7)

sini?',

in

7

that

is,

sin(5-iS)sin((7-7)sina8in((?'-5')+...=

;

*

See

Cremona's

Geometrie

t.

par

Dewulf,

p.

224.

The

bare

statement

that

if

one

triangle

an

be

circumscribed

abont

or

inscribed

in

a

circle and self

-conjugate

with

regard

to

a

second

circle,

n

infinity

f

such

triangles

an

be

drawn,

does

not

fullj

express

the

result.



WHICH

CUT

EACH

OTHER

AT

GIVEN

ANGLES.

61

or

since

sin

(G'-S ),

^miW-^F')

^m[r

-

G')

are

^sin^,

sinjS,

in

(7

respectively,

his

is

sin(jB

8)

8in((7

)

sin-4

sina

+

...=

0,

viz.

this is

sin

A

sin

a

sin

B

sin

yS

sin

C^

sin

7

_

sin(4-a)

^

sin(jB-/3)

^

sin(a-

7)

'

or as

this

may

also be be

written

1

1

2

cot-4

cota

cot

5-

cot

)8

cot(7

cot7~

Bat

assuming

this

equation

o

be

satisfied,

t does

not

appear

that

there

is

any

simple

expression

or

the

equation

of

the

line

through

the

three

centres

;

nor

would

it be

easy

to

transform

the

equations

o

as

to

have

this

line

for

one

of

the

axes.

The

case

in

question

(which

is

a

very

important

ne

from

its

connexion

with

Poincar^'s

theory

f

the

Fuchsian

functions)

may

be considered

independently.

Taking

the line of

centres

for the axis of

ar,and

writing

a,

/8,

7

for

the

abscissae

of

the

centres,

and

P,

Q^

R for

the

radii,

hen

the

equations

f the

circles

are

(;r-a)' /=P',

(x-/S)

/= 2',

and then if

the

pairs

of circles

cut at

the

anglesA^ B^

C

respectively,

e

have

Q'

+

2QBcofiA

+

B':={l3^y)\

R

+

2iZPcos5

+

P'

=

(7

-

a)',

P'

+

2P(2co8O+0 =(a-.i8)',

which

are

the

equationsonnecting

,

/8,

7,

P,

Q,

B.

See

the

figure.

It

is

to

be

remarked

in

regard

hereto

that

if

-4,

P,

G

are

used

to

denote

the

interior

angles

of

the

curvilinear

triangle

ABCy

then

the

angles

yA/3^

olBj^

0Ca

are

=7r

-4,

P,

G

respectively

whence

if

P,

Q,

B

were

used

to

denote

the

three

radii

taken

positively,

he

first

equation

ould

be

g*

+

2

(^P

cos^

4-

P'

=

(/8

7)*,



62

PROF.

CAYLET,

ON

THE

SYSTEM

OF

THBEE CIBCLES

as

aboTe

;

but the

other

two

equations

ould

be

i?-2i?Pcos5+P'

=

(7-a)',

P'-2P(2co8(7+(2

=

(a-^)';

hence,

in

order

that

the

equations

ay

be

as

above,

it

is

necessaiy

that

P

denote

the

radius

of the

circle,

entre

a,

taken

negatively

and it

in

fact

appears

that

in

a

limiting

ase

afterwards

considered

the

value

of

P

comes

out

negative.

Similarly

s

regards

the

curvilinear

triangleBC]

here

Ay

B{^B^)

and

C(= (T)

are

the

interior

angles

of

the

triangle;

and

the

radius

of

the

circle,

entre

o^,

must

be

regarded

s

negative.

Considering

^BÔba

given,

e

have

an

equation

etween

the

radii

P,

$,

R. In

fact

this is

at

once

obtained

in

the

irrational

form

\/(-3r)

VlÔ

+

Vl-^jÔ,

and

proceeding

o

rationalise

this,

e

obtain

-2V(rz)=r+z-x,

that

is

-V{(P'

+

2Pi?co85+i?)

(P'

+ 2PacosO+

^)}

=

P'

+

P(^

cos

(7+

^

cosP)

-

^5

cosX

Hence,squaring

nd

reducing,

e

find without

difficulty

0=

(2'i?sin ^

i2'P'sin'P+

PVsm'(7

+

2P'^5

(cos^

+

cosP

cos

O)

+

2P(^R

(cos^+

cos

(7

cos

-4)

+ 2PQR

(cos

C+

cos-4

cosP,

or

puttmg

herem

P, Q^

B^ r-

i

y-

,

*his

la

(cos^

+

cosBcosC cosP+cos(7cos4

^'^'^'

sini^ sin

0

'

sinCsin^

'

cosO+

cos-4cos

sin

^ sin

P

^(f,i7,5)'

Oj

and it

may

be

remarked

that

in

this

quadric

form

the

three

coefficients

are

each

less than

1,

or

each

greater

than

1,

according

s-4

+

P+

C7 7r,

r

-4+P+

C ir.

Suppose

r,

A-k-B-{-

C v\

the coefficients

are

here

B

cosX,

cos/i,

cosK,

the

form

is

(1,

1,1,

cosX,

cos

/A,

cosvXf

17,

?)*,

that

is

(f

+

17

cos

F

+

?

cos/i*)'

[if

smV

+

217?

cos\

+

?'

8in ,



r{8m'

WHICH

CUT EACH

OTHER

AT

aiVEN

ANGLES.

63

namely

thisis

,- ^ ,,

(

.

^cosX-

cos/Ltcosv)*

(f

+

17COBf+?cos/ia)'+

|i7Bmv+?

^^

1

(cos

COB

/14

cos

V\

')

^y

Jp

where the

last

term

is

=

-^

fsinV

sinV

(cosX

cos

ll

cos

v)*]

sm

V

^

' /

J

I

where

the

coefficient

n

{

}

is

=

1

cos'X

cosV

cosV

+ 2

cosX

cos/x

cos

v,

namely,

substituting

or

cosX,

cos/i,

cosv

their

values,

his

is

=

.

,

^

.

iT

At

(l-cosM-

cos'-B-

cos'

(7-2COS-4

cos5cos

0)'.

sm

-4

sm

5sm C

'

It

thus

appears

that

the

form

is

the

sum

of

three

squares,

and

is

thus

constantlyositive

it

therefore

only

vanishes for

imaginary

values of

the radii

;

or

the

case

does

not

arise for

any

real

figure.

Hence,

2'',

f

the

figure

be

real,

4

+

5+

(7 7r,

that is

the

sum

of the

angles

of

the

curvilinear

triangle

s

less

than

two

right

angles:

the

radii

are

connected

as

above

by

the

equation

(

cos

^+

cos

J?

cos

(7

cos

g

+

cos

g

cos

.4

'^'^

sin^sinO

'

sinOsin^

cos

G

+

cos

A

cos

B\

/sin

A

sin5

sin

G\

sin.4

sinjB

g\

/sin^

sin5

sInC\*_

in

which

form

the

three

coefficients

re

each

greater

than

1.

Eestoring

herein

f

,

i;,

f

and

regarding

hese

as

rectangular

coordinates,

he

equationepresents

a

real

cone

which

might

be

constructed

without

difficulty,

nd

then

taking

f,

i;,

f as

the coordinates

of

any

particularoint

n

the

conicalsurface

1

T^ ^ ^^

9in-4

sinJ5

sin

CJ'

^m

.

,

we

have

P, Q^

-B

=

^,

,

u-

Obviously,

oints

on

the

same

generating

ine

of

the

cone

give

values

of

P, Q^

R

which

difier

in

their

absolute

magnitudesnly,

their

ratios

being

the

same

:

the

original

quations

n

fact

remain

unaltered

when

P,

Q,

jS,

a,

/3,

7

are

each

affected

with

any

common

factor.



64

PROF.

CAYLEY,

ON

THE SYSTEM

OP

THREE

CIBCLES

Supposing

P, Qy

B

taken

so

as

to

satisfy

he

equation

n

question,

hen

taking

the radicals

'JiQ'

2

QR

cos

A

+

IT),

J{IP-{'2RPcosB'\-r)y

V(P*

+

2PCco8

0+^y

with

the

proper

signs

we

have

a

sum

=0,

and

these

give

the

values of

)8

7^

7

a,

a

^

respectively

and

the

construction

of

the

figure

ould be thus

completed.

I

look

at

the

question

rom

a

different

point

of

viewj

taking

Q^

E,

0

y

such

as

to

satisfy

he

first

equation

^'

4

2

5 cos

id

+

JB'=t

(/8

7)%

(that

i

starting

from the

two

circles

{x

-

^)'

+

y*= Q'f

{x-yY

+

y^^R^

which

cut

each other

at

a

given

angle-4),

then the

problem

is

to

find

a

circle

{x

a)'

y*

=

P*,

cutting

these

at

givenangles(7,

B

respectively

and

to

determine

the

coordinate

of

the

centre

a,

and

radius

P,

we

have th6

remarining

wo

equations

iZ''

2iilP

cos

P

+

P

=

(7

a) ,

P'+2P(2cosa+(2 =(a-/S) .

namely,

considering

,

P

as

the

coordinates

of

a

point

(io^

reference

to

the

foregoing

rigin

and

axes),

and

for

greater

clearness

writing

=

x,

P=y

we

have

y*

+

2yi?

cosP

+

i2

(x

-

7)'

0,

y'

+

2y(2cosa+(2*-(x-/S)'=a,

or a

these

may

be

written

{j

+

R

COB

By

-

(x

-

7)'

- R'

sin*P,

(y

4

0

cos

Cy

-

(x

-

P)*=

-

Q^

sin'G,

namely^

the

first

of

these

equations

denotes

a

rectangular

hyperbola,

oordinates

of

centre

(x

=

7,

y

=

P

cos

P),

transverse

semi-axes

=P8inP;

and

the

second

of

them

a

rectangular

yperbola

oordinates

of

centre

(x=/8,

y=-Qco8(7),

transverse

semi-axes

^QsinC:

as

similar and

similarly

situate

hyperbolas,

hese intersect in

two

points

nly

;

namely,

the

points

re

the

intersections

of either

of them

with

the

common

chord

2y

(R

cosP-

Q

cos

(7)

2

(7

^)

{x i

(7

+

/3)},+P'

Q'

=

0.

It

is

possible

o construct

a

circle

through

the

two

points

of

intersection,

nd

so

to

obtain

these

points

s

the

intersections



%



66

PROF.

CAYLEY,

ON

THE

SYSTEM

OF

THREE

CIRCLES

But

we

have

and

this

equation

hus

becomes

- =

16)8'

'W

{(cos^

cos

B

cos C)'

sin 5 sin'

G]

=16^8'

(^R\-

l+cosM+cos'^+cos'

(7+2cos^cos5cos

G).

We

have

therefore

2

{4/3*

[R

cos5-

e

cos

C)'}

+

{4/9*

5

cos5+

Q

cos

C)

-

{R

cos5-

^

cos

(7)

[R -

Q')}

=

4/8

Q5

V{-

(

1

-

cosM

-

cos*J? cos*

(7-

2

cos^ cos5

cos

C)

}

4(5cos5-

^cos(7)y

+

ii *-

G'

=

4i8x,

or

completing

he

reduction

by

the

substitution

of

the

value

of

4/3*,

his

is

y

{(^

sin (7-f

^

sin'5)

2Qfi(cos^

+

cos5cos

G)}

^

QR{Q

(cos5+

cos

G

cos^)

+

R

(cos

(7+

cos^

cos

5)}

=

4/SQ5V{-

(1-

cosM-

cos'J?-

cos'

(7-

2

cos^

cos5cos(7)},

viz.

we

have

thus

two

values

of

the

radius

y

(=P)

;

and

to

each

of

these

there

corresponds

single

value

of

the

abscissa

x,

given

by

4)8x

^'-

^

+ 2(5co3J5-.

QcosC)y.

The

two values

become

equal

if

-4

4

jB

0=7r;

in

this

case

the

three

circles

meet

in

a

pair

of

points

(x^,

y,),

(x,,

y^).

In

fact,

riting

4

+

5+

C=7r,

and

thence

C08-4

=

cos

(5+

(7),

cos

J?

cos

(7

+

sinjB

sin

(7,

c.,

we

find

{G

sin'0+

2^^

(cos^+

cosJ?cos

G)

+

iZ'

sin'P}

\^QR[Q

(cosjB

cos

C

cos-4)

R

(cos

C+

cos^

cos5)}

0,

that

is

(Q

sin

(7+5

sin

J?)'y

QR

(^

sin

0+

R

sin5)

sin-4

=

0,

or

throwing

ut

the factor

[QAnG-^-R

sin

J?)

this

is

[Q

sin

(7+

R

sin

5)

y

+

$5

sin^

=

0,



WHICH

CUT

ACH

OTHER

AT

GIVEN

ANGLES.

67

and

we

then

have

-

2

(

COB^

-QcmO)

QB

fimA}.

The

term

in

{ }

is

here

.

B?{AnB)

\-B?Q{

8in(7-2

8in^co8J5)

+

5^

(-

8in5+

2

sin^

cos

C)

+

i^[-AnC),

which

Is

=

i?( sin5)

+

5'Q

(-

sin

(7+

2

sin

J?

C08-4)

+

- '(

8in5-2

8in(7cos-4)

=

(5*

+

0'

+

2i?(2

cos^)

[R

AnB^Q

sin

(7),

4^(^sin5-^sin(7),

or

finally

'^ ^sin^H-

sinC

_/3(Jg8in^-gsing)

^

^sinjB+QsinC?

*

In

these

equations

y,

x

should be

replacedby

P,

a

respectively;

nd

in

obtaining

hem

it

was

assumed

that

7

=

-

/3;

restoring

he

general

values

of

/8,

the

equations

become

p

^jRsin^

iZsinJS+Csina*

^

i^o..A_4(^-7)(^8mg- (g8ing)

-i(/S+7)

i23in +(2sinO

'

F2



68

PROF.

CAYLEY,

ON THE SYSTEM

OF

THREE CIRCLES.

viz.

this last

equation

ecomes

*

EsinB-{-QainG

'

or

say

a

{B

sin5+

Q

sin

C)

-

I3B

smB-yQ

sin

(7=

0,

y^hich

by

means

of

the

first

equation

ecomes

a-^sin^

i8iisinjB+7(28ina=0.

It

thus

appears

that

the

two

equations

re

sin^ BinJ5 sin

(7

T

^

Q

^

B

=

0,

asin^

8

H\nB

y inC

^

-p-

* - -+

S-=^'

viz.

these

equations,

herein

-4

+

-B+C7=7r,

belong

to

the

case

where

the three circles

intersect

in the

same

pair

of

points;

ence if

the

coordinates

x, y

refer

to

the

points

f

intersection of the three

circles,

e

have

simultaneously

he

equations

f

the

three

circles,

nd

the three

equations

hich

determine

the

angles

at

which

they

intersect,

iz.

we

have

the

six

equations

(aj-a)'

y*=PV

Q'

+

B'-\-2QBcosA

=

{^-'

y)%

{x-j3Y+y'^Q\

fi'

+

P'

+

25Pcos5

=

(7-a)',

(aj-

yy'\-y'=-B\

P +

(? 2P(?

cos

C

=

(a

-/3)V

viz.

from these six

equations,

ith

the

condition

A

+

B-\-

G=irj

it

must

be

possible

o

deduce

the

last-mentioned

pair

of

equations.

In the

general

case,

where

A-\-B-\-

G ir,

and the

three

circles

do

not

meet

in

a

point,

then

taking

the

circles

(x

/8)*

y

=

Q^

(x

7)*

-f

y^^B*

to

be circles

cutting

ach

other

at

the

angleA,

or, what is the

same

thing,

he values

Qy

By

/8,

to

be

such

as

to

satisfy

he

relation

^ +

5'

+

2Q^cos^

=

(i8-7)%-

the

two

equations

or the

determination

of the

abscissa

of

the

centre

a,

and

the

radius

P

of

the

remaining

circle

give,

by

what

precedes



MR.

DAWSON,

A

THEOREM IN

HIGHER

ALaEBRA.

69

2

{(yS

7)

{B

COS

B-

Q

cos

(7)*}P

+

{(/8-7)'(^co85+

CcosC)-

(Bcos5-

gco8C)(ii'-

C'j}

= 2(/8-7)C^V{-(l-co8*^-cos'5-cos (7-2cos^co85cos(7)}

4(i2cosB-(2oo8(7)P+(iZ'-

e2')

(i8-7)(2a-i8-7),

viz.

we

have thus the

two

circles

(x

a)*

+

y*

=

P*,

each

of

them

cutting

he

circles

(x-/8)*+y*=

Q*,

and

(x

-

7)*

+

y*

jB'

at

the

angles

C^

B

respectively.

NOTE

ON A THEOREM IN HIGHER ALGEBRA.

By

H. O,

Dawson,

B,A.

The

theorem

concerns

the

equivalence

f

the

operators

and

o^2a,^^

3a.A

c,

when

applied

o

a

function

of the

quantities

4

,

A^^

A^...A^j

where the

transformation

x=pX'^qYy

y=p'X

+

qY

trans-orms

the

binary

form

iaja.,..a)

x.

yY

into

the form

This

equivalence,

hicU

is

of

importance

n

the

theory

of

the

covariants of

the

form,

is

proved

somewhat

tediously

n

Faa-de-Bruno's

Theorie

des

formes

btnaires^

p. 189,

the

equivalence

aving

been

previously

tated in

Section

77.

The

proof

now

given

seems

interesting,

ot

only

as

being

expeditious,

ut

also

as

pointing

out

very

precisely

he

raison

(TStre of

the

equivalence

n

question,amely,

that it

is

a

necessary consequence

of the linear

character of

the

transformation.

The

proof

in

question

s

as

follows

:

Let

us

suppose

that the

transformation

x={p^Xp')X+{q-^\q')Y,

y=/J:+}'r

is

made,

then

we

get

a

new

form

whose

coefficients

re

dp

dq

,

X

being

supposed

o

be

indefinitely

mall.



70

MR.

DAWSON,

NOTE

ON

A THEOREM

Now,

owing

to

the

linear

character

of

the

transformation,

we

might

have

proceeded

in

a

different

manner,

thus^

we

might

have

put

fl5

SB

u

+

Xv,

y

=

t?,

and

afterwards

transform

by

the

substitution

The

first

of

these

two

transformations

gives

s

a

form

whose

coefficients

re

a^,

a^

+

2\a^j

a

+

3\a^j

c.,

supposing

*

to

be

inappreciable,

hilst

the

second

^ives

-4/,

k/,

..,^/)

X,

F)*,

where

A^\

A^^...yA^

are

just

the

same

functions

of

a^,

a^

+

2Xa5,

a^

+

3Xag,

c.,

that

-4^,

^...A^

re

of

a^,

a^...a^.

But,

remembering

ur

former

view

of

the

transformation

we

have the theorem

in

question,

or

where

y|

+

j'|=8^.

The

proof

is

easily

pplied

o

other

forms;

take

for

an

example

the

ternary

n'

and

write

it

as

follows

:

Suppose

that

when

we

make

the

transformation

we

obtain

{A)

(Z,

F,

Z)\



IN

HTQHEK ALaEBRA.

71

Let

as

now

make

the tranformatlon

=

(X,

+

Â,

+

f,X,)

+

(^,

+

Â*.

Ât.)

+

(v, 0v,

+

ipv,),

we

thus

obtain

a

form

whose coefficients

re

where

g;

X.

A

+

^,^

^,^

S.= c.;

again,

the transformation

could

have

been

performed

in

another

order,amely,

by

first

using

the

substitution

and

afterwards

putting

Now

if

we

make

the

first

transformation,

e

easily

ee

that

the

coefficient

a^â

changes

o

[This

is

easily

obtained,

for

0

and

^

are

of

course

course

supposed

o

be

very

small

thoughperfectlyrbitrary

we

are

therefore at

liberty

o

neglect

all

terms

higher

than

the

first

(in

0

and

0)

of such

a term

as

{U+0V+4 W)'^.

Eesuming,

if

we

perform

the second

transformation

we

obt iin

a

form

[A')

[XYZfy

whose

coefficients

re

the

same

functions

of the

quantities

that

A^, -4,0,..

c.

are

of

a^, r,

c.,

and therefore

dA

where the

2

applies

o

all such

values

of

Up^^

as

can

enter

into

Ap^tr*



72

MR.

DAWSON,

A THEOBEM

IN

HIGHEB

ALGEBRA,

But

we

obtained

before

tbe

equation

HeQoe

we

have

an

equivalence

etwen

the

operators

and

between

for

this

reason,

that

6

and

^

are

perfectly

rbitrary.

By writing

he form in powers of y^

we can

obtain

two

operators

hich

will be

equivalent

o

^

d

d

d

^

^

d

d

d

similarly

e

can

escpress

the

operators

^

d

d

d

-,

d

d

d

in

terms

of

the

coefficients.

The

invariants

of

the

form

satisfying

he

six

equations

V

=

0,

8/

=

0,

8/'

0,

8/'

0,

8/''

0,

8/'^=

are

the-common

solutions

of

the

equations

hus

found.

If

we

take

the

case

of

the

quadric

orm

(a,J,

c,/,

^

h)

{ocy^Y

our

equations

re

d

.d

^

d

d

^^d

d

Interchange

?,

y

and

we

get

Interchange

,

ssj

'd/^^dh-'^^M^'^dh^^^di^'d^

mi

BQ

on

for

other

forms.

Qhivrt'B

ollege,

Cambridge,

Vtt/i/0,

1887.



PROF.

CAYLET,

ON

SYSTEMS

OP

RAYS.

75

the

ra;^8

at

rightangles.Evidently

,

/8,

=

8

F,

8^F,

S^V

reBpectiveljy

nd the

function

V

satisfiesthe

partial

ifferential

equation

(8.F)'

+

(S,F)'

(8.F)'

1.

Hamilton

in effect considers

only

systems

of

rays

of

the

form in

question,

iz. those which

are

the normals of

a

surface

(or,

what is

the

same

thing,

the normals of

a

system

of

parallel

urfaces),

nd

it

is

such

a

system

which issaid

to

have

the

characteristic

function

V.

It

is

shown that

a

system

of

rays

originally

f

this kind

remains

a

system

of

this

kind

after

any

number of

reflexions

(or

ordinary

refractions)

in

particular

f the

rays

originally

manate

from

a

point,

hen,

after any

number

of

reflexions

at

mirrors of any form what-ver,

they

are

a

system

of

rays

cut

at

rightanglesby

a

surface.

And

moreover,

there

is

given

for

the surface

a

simple

construction,

iz.

starting

rom

any

surface

which

cuts

the

rays

at

rightangles,

nd

measuring

off

on

the

path

of

each

ray

(as

reflected

at

the

mirror

or

succession

of

mirrors)

ne

and

the

same

arbitrary

istance,

e

have

a

set

of

points

forming

surface

which

cuts

at

right

angles

the

system

of

rays

as

reflected

at

the mirror

or

last

of

the

mirrors.

The

ray-systems

considered

by

Hamilton

are

thus the

normals

of

a

surface F

c

=

0,

and

a

large

part

of

the

properties

f

the

system

are

thus

included

under the known

theory

of

the

normals

of

a

surface

;

it

may

be

remarked

that

the

analytical

ormulaB

are

somewhat

simplified

y

the

circum-tance

that

V

instead

of

being

(as

usual)

an

arbitrary

function

of

(a;,

y,

z)

is

a

function

satisfying

he

partial

differential

equation

(8 F)'+

(8^F)*+

Sr)*

=

l.

In

par-

ticular

we

have

the theorem that each

ray

is

intersected

by

two

consecutive

rays

in

foci

which

are

the

centres

of

curva-ure

of

the

normal

surface

;

the

intersecting

ays

are

rays

proceeding

rom the

curves

of

curvature

of

the

normal

surface,

c.

There

are

other

properties

asily

educible

from,

but

not

actually

ncluded

in,

the

theory

of the normals

;

for

instance,

the

intersecting

ays

aforesaid

are

rays

proceeding

from certain

curves

on

the

mirror,

analogous

o,

but

which

obviously

re

not,

the

curves

of

curvature

of the

mirror.

The natural mode of

treatment

of this

part

of the

theory

s

to

regard

the

rays

as

proceeding

ot

from the

normal

surface,

but

from

the

mirror,

and

(by

an

investigation

erfectly

analogous

o

that

for

the normals of

a

surface)

o

enquire

s

to

the

intersection

of

the

ray

by

rays

proceeding

rom

con-ecutive

points

f

the

mirror

;

it

would

thus

appear

that

there



76

PBOP.

CAYLEY,

ON

SYSTEMS

OP

RAYS.

are

on

the

mirror

two

directions,

uch

that

proceeding

long

either of

them

to

a

consecutive

point,

he

ray

from

the

original

oint

is

intersected

by

the

ray

from

the

consecutive

point,

ut

that these

directions

are

not

in

general

t

right

angles,

c.

But

in

regard

to

such

an

investigation,

he

restriction

introduced

by

the

Hamiltonian

theory

is

altogether

nne-essary

;

there

is

no

occasion

to

consider

the

rays

which

proceed

rom

the

several

points

f

the

pairror

as

being

rays

which

are

the

normals of

a

surface,

nd

the

question

s

con-idered

from

the

more

general

oint

of

view

as

well

by

Malus

in his

Th^orie de

la

Double

Befraction,

c., Paris,

1810,

as

more

recently

by

Eummer

in

the

Memoir

AUgemeine

Theorie

der

gradhnigentrahlensysteme,

relUy

t.

57

(1860),

pp.

189

230y

viz.

we

have

in

Kummer

a

surface of

any

form

whatever

(defined

according

o

the

Gaussian

theory,

a;,

y,

z

S'ven

unctions of

the

arbitrary

arameters

m,

r),

and

from

e

several

points

hereof

rays

proceeding

ccording

o

any

law

whatever,

viz. the

cosine-inclinations

a,

/S,

7

(or

as

Eummer

writes them

f

,

17,

g) being

given

functions

(such

of

course

that

a*

+

^8

+

7*

=

1)

of the

same

parameters

m,

v.

It

may

be remarked: 1* that

Kummer,

while

considering

he

simplifications

f the

general

theory

presenting

hemselves

in

the

case

where the

rays

are

normals

of

the

surface,

oes

not

specifically

onsider

the

case

where,

not

being

such

normals,

they

are

(as

in

the

Hamiltonian

theory)

ormals of

a

surface.

2^

that

some

interesting

nvestigations

n

regard

to

the

shortest distances between

consecutive

rays,

while

naturally

connectinghemselves,

ith

the

normals of

the

surface,

r

with that of the rays which

are

normals of another

surface,

do

not

properly

belong

to

the

AUgemeine

Theorie

of

a

congruence,

which

is

independent

f

the

motion

of

rect-

angularity.

It

has

been

already

remarked

that

the

system

may

be

looked

at

in

the

two

ways

1'

and

2*,

and itis in the

former

of

these

that

the

question

s

considered

by

Kummer;

it is

interesting

o

work

out

part

of

the

theory

in the

latter

of the

two

ways.

Taking

X^

Y,

Z

as current

coordinates,

e

have,

for

a

line

through

the

point(a;,

,

is),

he

equations

Z,

F,

Z=aj

+

a/ ,

y

+

^p,

+

7p;

a,

/8,

7

are

functions of

(a?,

,

z\ satisfyingdentically

he

equation

*

+

i8^

7*

=

1

(and

therefore

the

derived

equations

in

regard

to

(c,

y,

z

respectively);

nd also

satisfying

he



PBOP.

CATLET,

ON

SYSTEMS OP BATS.

77

eqaations

(a8.

i88^

7S.)a

0,

(aS.

)88^

7SJ

/S

=

0,

(aS.+

i8S^

7S.)7-0.

It should be remarked

that

if

these

equations

ere

not

satisfied,

hen

instead

of

a

congruence

there

would be

a

complex,

or

triply

nfinite

system

of

lines,

iz.

to

the

several

points

of

space

(x^

y,

z)

there would

correspond

ines

X,

F,

Z^

x

+

OLpy

y

+

i p,

S

4

7p

as

above,

which

lines

would

not

reduce themselves

to

a

doubly

infinite

system.

Suppose

that

the

line

through

the

point

y

y^

z

is

met

by

the

line

through

a

consecutive

point

{x

+

dx^

y

+

dt/jZ-^z)

;

then,

if

X, F,

Z

refer

to

the

point

of

intersection

of the

two

lines,

e

have

dx

+

adp

+

pdOf

dt/'\rl3dppd/S^

dZ'\-ydp

pdj=^Oi

and

consequently

dxy

duj

d

dy, dp,

fi

dzy

dy,

7

as

a

relation

connecting

he

increments

dx,

dy^

dz,

in order

that the

lines

may

intersect,

iz. this is

a

quadric

relation

(^ydxydy,

dzy

=

0

between the

increments.

In

the

case

of

a

complex

this

equationrepresents

cone

(passing

evidently

through

the

line dx

:

dy

:

dz^a:

fi

:

7),

but

in

the

case

of

a

congruence

the

cone

must

break

up

into

a

pair

of

planes

intersecting

n

the line in

quisstion

x

:

dy

:

dz

=sai

0

i

y.

To

verify

h

posteriori

hat

this

is

so,

observe that

the

differential

equations

atisfied

by

a,

/8,

give

as

above

8,7- A

8.a-S.7,

S^-S,a

proportional

o

a,

^,

7,

or

say

=

2Aa,

2A)8,

2A7

;

and

it

hence

follows

that

the

differentials

a,

S/3.By

can

be

expressed

n

the

forms

da

adx

+

hdy

+

gdz

+

JcdSdz ydy),

dl3^hdx+

bdy

+/dz

'\-

k

(y

dx

-

adz),

dy

gdx+

fdy

+

cdz

+

k(ady

-fidx),

where

0

=

aa

+

A^S

+

gy,

0

=

Aa+ i8+/7,

0

=

flra+//3

C7,

.



(

80

)

A NEW

METHOD FOR

THE

GRAPHICAL

REPRESENTATION

OF COMPLEX

QUANTITIES.

By

jr.Brill.

1. As

it

is

now

more

than

eighty

years

since

Argand

introduced

the

well-known

method

for the

graphical

epresen*

tation

of

complex

quantities,

nd

the

principle

f

duality

as

long

been

recognized

y

geometers,

it

seems

strange

that

it

should

have

occun'cd

to

no

one

to

apply

the

same

idea

to

tangential

oordinates

;

in

other words

to

construct

a

diagram

in which

complex

quantities

hould

be

represented

y

lines

instead of

points.

It

might

be

anticipated

hat the theorems

obtained

by

this

method

would

be

none

other

than

the

polar

reciprocals

f theorems

similarly

btained

by

the older

method

;

however,

since

metrical

properties

flFer

considerable

diflSculty

o

reciprocation,

t

might

be

advisable

to

have

a

method which

would

yield

he

reciprocal

roperties

irectly*

I

have

endeavoured

to

supply

uch

a

method

in the

following

paper.

It

will

be

founa

that the

theory

is

not

an

exact

analogue

of that in

which

complexquantities

re

represented

by

points.

This

is

as

might

be

expected,

or

itis well

known

that

complete

duality

does

not

exist

in

the

planimetry

f

homaloidal

space.

2.

Before

proceeding

o

develope

he

theory

e

will

prove

a

theorem

which will be of

use

in

the

course

of the

investi-ation.

0

is

a

fixed

point,

nd

a

straight

ine is

drawn

through

meeting

n

fixed

lines

in

the

points

^,r,,

...y

r^.

A

pomt

R

is

taken

on

the

line

through

0,

so

that

OB

ft^ T, ^- ^ ^-

The

locus

of

S will

be

a

straight

ine.

To

prove

this take

two

rectangular

xes

through

0,

and

let

the

equations

f the

given

lines

referred

to

these

axes

be

Mjic

+

Vjy

-

1

=

0,

u^x

+

1?^

1

=

0.

..tf^sTj



MR.

BRILL,

ON

COMPLEX

QUANTITIES.

81

Then,

if B

be

the

angle

made

with the

axis

of

x

by

the

line

through

0,

we

have

TT

=

u.

cos5

+

V.

sin

0.

Tr

=

^i

cos^

+

v,

sin

tf,

-pr

=

M

cos^

+

sin

0,

Therefore

the

equation

f the locus

of

B

is

'^ ^'^' * '^'^'*=ffl,(^,cosg+t^,8ing

r

Swim

.

Smv

^

.

We

shall call

this

the

mean

line

with

respect

to

0

of

the

n

given

lines

for

multiples

j, wi

...,

?n^.

3.

Let

the

equation

f

a

straight

ine

be

given

in

the

form

ux

+

vy l =0,

where

u

and

v

are

the

reciprocals

f

the

intercepts

n

the

axes.

We

shall denote

the

position

of this

line

by

the

expression

4-

iv,

or

if

we

desire

to

use

a

single

ymbol

by

w.

This

being

premised,

e

see

that

the

line at

infinity

ill

be

represented

y

zero,

and that

any

line

through

the

origin

ill be

represented

y

an

infinite

complex.

Beal

quantities

ill

represent

lines

parallel

o

the

axis of

y^

and

purely

imaginary

quantities

ill

represent

lines

parallel

to

the axis of

x.

Further,

ny line

parallel

o

the

given

ne

will be

represented

y

a

real

mjiltiple

f the

expression

which

denotes

the

given

line.

4.

Let

there

be two

given

straight

ines whose

equations

Bre

t aj

+

vy

1

=

0

and

ux

+

v'y

1

=

0.

Consider

the

line

m

{ux

+

vy

-

1)

+

n

[u'x v'y 1)

0,

or as

it may

be

written

mu^nu'

mv

+

nv*

x^

y-

1

=

0.

Let this

be

equivalent

o

Kb

+

T^

1

=

0,

then

we

have

{m

+

n)U=mu

+

nu*

and

(m

+

)

r

twv

+

v'.

VOL. XTII.

G



82

MS.

BRILL,

A NEW METHOD

FOB

THE

GRAPHICAL

Hence,

if

W=V+iVj

w=u+tv^

and

w'=u'+w'^

we

have

(wt n)

W^mw-{-nw\

The line

W

is what

we

have

calledthe

mean

line of

the

lines

w

and

w'

for

multiples

and

n.

If

we

make

m

=

n,

then

we

have

2W=w

+

xo\

In

this

case

W

will

coincide

with the harmonic

polar

of

the

origin

with

respect

o

the

lines

w

and w'. Thus

to

obtain

the

line

which is the

sum

of

two

given

lines,

e

have

to

draw

the

harmonic

polar

f the

origin

ith

respect

to

them,

and then

to

draw

a

line

parallel

o

this

and

at

one-half the distance

from

the

origin.

Thus

the

idea of

the

mean

line

furnishes

us

with

an

interpretation

f

the addition

of

lines,

ust

as.

the

idea

of the

mean

centre

furnishes

us

with

an

interpretation

of the

addition of

points.

The

actual

position

f the

mean

Jine,

however,

depends

upon

the

position

f

the

origin,

o

ths^t

the

jinalogy

s

not

quite

complete.

If

we

write

??i

4

n

=

0,

then

TV

becomes

a

line

through

the

origin,

pd

all

that

we

can

say

is

that

w

vS

is

some

line

{parallel

o this.

It

will

therefore

be

necessary

to

investigate

he

case

oi

'{jo

yi

separately.

he

equation

f

this

line

ia

(u

w')

; +

(v v')

1

=?

0,

This is

ohvioqsly

arallel

o

(t*

w')

?

+

(v

v')

=

0,

the

line

joining

the

origin

to

the intersection

of

vo

and

v)\

In fact

this

latter line coincides

with

TT.

Further,

the

equation

f

to

to'

may

be

written

iB

+

vy

1

[yix v'y)

0,

which shews

that it

passes

through

the

intersection

of

w

with

H

Une

drawn

through

the

origin

parallel

o

w\

Thus,

to

construct

the line

v

w',

we

draw

through

the

origin

line

parallel

o

w\

and

through

the

intersection

of this

with

w we

draw

a

line

parallel

o

that

joining

he

origin

o

the

intersex*

tion

of

w

and

w\

.

If

TF be the

mean

of

the lines

^j,

w^,

...,

^

for

multiples

m^

w

.,.,

i^,

then,by

proceeding

s

in the

early

part

of

this

article,

e

should

obtain

(Wj

+

w,

+

. .

.+

?w

J

Tr=

m^^

+

m^^

+.

.

.+

Vijio^^

6,

Let

the

straight

ine

t aj

+

t?y

1

=0

cut

the axis

of

x

fit

the

point

-4,

and

that

of

y

at

the

point

5;

and let

OA^a

l^nd

QB ^b,

From

Q

draw

OL

perpendicular

o

-45.

Let



BEPKESENTATION

OF

COMPLEX

QUANTITIES.

83

OL=p^

and

let

0

be

the

angle

which

OD

makes

with

the

axis of

X.

Then

we

have

1

CO8

0

,

1

sin^

tt=

-

=

,

and

v=T

=

.

a

p

^

op

Therefore

1

.

*

w

=

u

+

iv=i-

(cos5

+

isind)

P

V

This

formula

will

enable

us

to

discuss

the

question

f the

multiplication

f

two

lines.

For

let

wlc

=

WjW^

where

w,

fff^j^

denote

lines

and

c

denotes

some

length.

Substituting

for

Wj

w^

and

w^

their values

as

given

by

the above

formula

we

obtain

cosg

+

tsing

_

cos

0^

+

i

sin

0^

cos^^+

i

sinS,

^

cos

{0^ gj

+

i

sin

{0,

0,)

Hence,

we

have

co8g^cos(g^-f

,)

.

sing

_

sin

(g^ g,)

^

PiPt

*

cj

'

.

p,p,

'

from

which itfollows

that

^-P\P%

**^d

0^0^-\-0^.

In

a

similar

manner

it

can

be

proved

that

if

vf^vi^^

then

P*

'=^PxPt

and

2g

=

g,

+

g,.

6. As

a

first

example

of

the

utility

f the method

we

will

investigate

he

analogue

of the well-known

proof

by

Argand's

method

of uc. I.

47.

Taking

the

figure

f

the

last

article

produce

BO to

B, making

OJS'=

OB.

Then

the

line AB

will be

represented

y \\a-\-i\h^

nd the

line

AB^

by

\\a

ilh.

Further,

rom

the

last

article

we

have

1

t

_

*

a

0

p

Also,

the

perpendicular

rom

0

on

AB

is

equal

in

length

to

that

from

0

on

AB^

and

the

two

perpendiculars

ake

equal

angles

with

the

axis

of

x

on

opposite

ides

of

it;

therefore

ah

p

'

G2



84

MB.

BRILL,

A

NEW

METHOD FOR THE

GRAPHICAL

Multiplyingogether

hese two

equations

e

obtain

1

+

1

=

1

which expresses

a

well-known

properly

of the

right-angled

triangle.

7.

We

will

now

proceed

to

develope

some

metrical

properties

onnected

with

the

theory

of

the

mean

line.

JLet

w

be

the

mean

of

the

lines

w^

and

w^

for

multiples

971

and

fij

then

we

have

{m

+

n)w^

mw^

+

nw^.

This

is

equivalent

o

the

two

metrical

relations

,

.cosS

cos^,

cos

9,

(m

+

n)

=

i

-+n

=

P

Pi

Pf

,

V

sin^

sin

0.

sin

6^

and

(m

+

n)

=

m

*

+

n

'

.

P Pi Pf

From these

we

easily

educe

the

two

following

p

Pi

p,

PiPf

and

^^^^^

=

-

cos(5

^,)

+

-

cos(^

0X

P Pi

'

Pf

*

Let

AXj

ABy

^40

be

the lines

respectively

enoted

by

to,

w^j w^.

From 0

draw

OL, OM^

01s

respectively

erpen-icular

to

AX,

ABj

AG.

Then,

if

0

lie within the

angle

BAG

or

the

vertically

pposite

ngle,

e

have

Im

+

)

Iff?

^

IP?

^

mn

t%

a

n

but}

if

0 lie

without

both these

angles,

e

have

(m

+

n)'_

m*

^

r?

^^

mn

^.^

or

^'OW'^ON^'^

OMTON^^

Through

L

draw

LH and

LK

respectivelyerpendicular

to

OM

and ON. The

second of

the

above

relationsbecomes

OH

OK



86

MR.

BRILL,

A

NEW

METHOD FOR

THE

GRAPHICAL

If

we

split

his

into its

real

and

imaginary

parts,

multiply

the

first

by

cosr^

and

the

second

by

sinr^

and

add,

we

have

^ '+^=^;co8r(^.-5)+r^'^cos{(r-l)(

.-^}

V

K

' '

'

Pi

P

+...+

cosr

(5,-^).

p;

It

would

be

troublesome

to

enumerate

all the

cases

of

this

that

might

arise,

o we

will

content

ourselves

with

one

particular

ase.

Let

0

lie

without

both

the

angle

BA

O

and

the

vertically

pposite

ngle,

and let

m

and

n

be

both

of

the

same

sign

so

that AX liesbetween AB and A

C.

The above

property

becomes

(m + n)'

w'

(

-DA

V)

H-

.

.

.+

-^^r

cos

{r

(7-4Z}.

This is

the

generalizednalogue

of

a

property

of

the

triangle

published

by

me

in

the

Educational

Times for

October

1885

(No.

8290).

9.

Let

?,

and

w^

denote

the

lines

AB

and

-4(7,

the

origin

0

lying

withm

the

angle

BAG. Draw

AX the

harmonic

polar

of

0

with

the

respect

to

the

lines

AB snidAG,

On

the

opposite

ide of 0

from

AX^

and

at

a

distance

from

0

equal

to

one-half

of that of AX from

0,

draw

a

line BG.

Then,

if

this

line be

denoted

by

w^,

we

have

w^

+

w^

+

w^

=

0.

This is

equivalent

o

the

two

metrical

relations

lind

cos

5,

cos^, cosS,

^

i-+ 2.+

5=0

A

P.

Pz

8in^^sln^_^

lnj93^^

P^ P.

P.

'

Fi F,

Pz



88

MR.

BRILL,

A

NEW

METHOD FOR

THE

GRAPHICAL

Then the

expressionjw^ s^/w^

s

equal

to

f

C08(^,-

,)

-^

C08,(^.-

j

+

.-If'

in(^,-,)

-^

8in(^,-d,

Stf

Sfa VSfi Sfa

J

and thereforeitsmodulus

is

Hence,

we

have

the

following

heorem

:

Let

ABC

and DEF

be

two

triangles

aving

a common

centroid

0,

and let

OL, OM,

ON,

OP,

OQ,

OR

be

the

respective

erpendiculars

rom

0

on

BC,

CA,

AB, EF,

FD

DE'y

then

we

have

fON\'

(0L\^ ÔN.OL

,o ,

fqL\\/OM^.*

ÔL.OM

111

Hopj'^vooj'ôp.oe'''''^^~^^--oP'oE^^'o^'

11. Let

ABC

be

a

triangle.

ake

a

point

0

within

the

triangle,

oin

OA, OB,

00,

and draw

OB,

OE,

Oi^

respec-

TIT

tivelyerpendicular

o

BG, GA,

AB.

Through

0

draw

OX

parallel

o

AB

to

meet

GA

in

X,

and

through

X

draw

JfiV

parallel

o

OA.

Through

0 draw

OY

parallel

o

J5(7

to

meet

AB

in

F,

and

through

Y draw

-A/i

parallel

o

OB.

^Through

0

draw OZ

parallel

o

GA

to

meet BG in

-2',

nd



90

MB

BBILL|

A NEW

METHOD

FOR

THE

GRAPHICAL

Let

MNj

NL,

LM

meet

AB^

BCj

GA

respectively

n

X\ Y\

Z\

Then

this relation

becomes

cos(Zr^-ZZr')

cos(Jf^^X~J/ZZ^

Olf.OP

*

OE\OQ

cos(NX'Y^NYX')

_

1

OF'.OB

OROQ.OR^

Similarly

e

may

deduce from

the

second

formula

the

relation

co^{AY Y-^AZ Z)

co^(BZ Z^

BX X)

OE.OF.OP

*

OF.OD.OQ

co (GX X^CY Y)^

OD.OE.OB

OROQ.OB^

where

X .

Y'\

Z

are

the

respective

ntersections

of

MN^

NL,

LM

with

BG,

GA,

AB.

13.

We

could obtain

analogues

of

all

the

properties

f

rectilinear

figures

btained

in

my

paper

on

Argand's

method.*

We

will,owever,only

consider

one more

instance^

iz.

the

interpretation

f

the

formula

Let

ABG

be

a

triangle

nd

0

a

point

within

it.

Let

MN be

the

harmonic

polar

of

0

with

respect

to

GA

and

AB,

NL

that

of

0

with

respect

to

AB and

BG^

and LM that of O

with

respect

to

BG and

GA.

Draw

CD, OE, OF, OP^ OQ^

OR

respectively

erpendicular

o

BG^

GA^

ABj

MN^

NL^

LM.

Then

we

have

cos(LBG'-LGB}

cob(MGA^MAG)

cos(NAB-NBAy

OD\OP

*

OE\OQ

*

OF\OB

cosiMGA'-NBA)

cos(NAB-LGB)

coB(LBG-MAGy

*

OE.OF.OP

*

OF.OD.OQ

*

OD.OE.OB

*

14.

Let

0

be

the

origin

nd

A

a

fixed

point

distant

2a

from

0.

On

OA

as

diameter

describe

a

circle,

nd

through

A

draw

a

line

AB

cutting

the

circlein L.

Draw

a

tangent

LM

to

the

circle,

oin

OL^

and draw OM

perpendicular

Messenger

of

AfathenuUicSf

ol.

xvi.,

pp.

8-20,



92

MR.

BRILL,

A

NEW

METHOD FOR THE

GRAPHICAL

on

the

tangent

at

L for

our

initial

direction.

We

have

ON'=:OM.OM',

and

the

angle

MOM'^^.MON.

Thus,

if

(?Jf=2c,

and

LL' be

represented

y

w^^

then

LM' is

represented

y

2cw\

Draw another hne

LL

throughi,

and

draw

ON'

per-endicular

to

LL .

Let

L M

be

the

tangent

at

L\

and

OM

the

perpendicular

n

it

from

the

origin.

Join

LL ^

and draw

UK

perpendicular

o

it.

Then

we

have

ON'^OM.OM'

and

ON'^^OM.OM ,

and

therefore

0N\ ON

=

0M\

OM'.

OM

=

0M\ OK'

;

that

is,

ON. ON'

=

OM. OK.

Further,

^

bisects

the

angle

M'OM ,

and therefore

2.MOK=:MOM'

+

MOM

=

2(MON-i-

MON')\

that

is,

MOK^

MON+

MON'.

Thus,

if

LL

be

representedy

w^,

and

LL

by

w

then

iX

is

represented

y

2cw^w^.

15.

As

a

first

example

of

the method

of

interpretation

developed

n

the

last

article

we

will

take the

formula

for the

product

f two

polynomials.

This

will

be

easily

een

to

yield

the

following

heorem

:

0 and

L

are

two

fixed

points

on

the

circumference of

a

circle.

A

series

of

m

fixed

points

s taken

on

the

circum-erence

of

the

same

circle,

nd

a

second

series

containing

n

points

s also

taken.

Let

LM

be

the

mean

line

for

equal

multiples

ith

respect

to

0 of the lines

joining

to

the

m

points,

nd let LN

be

the

mean

line for

equal

multiples

with

respect

to

0

of the

lines

joining

L

to

'the

n

points.

Let

LM and

LN

meet

the

circle

in

M

and N

respectively

then

MN

is

the

mean

line for

equalmultiples

ith

respect

to

0

of the

mn

linesthat

can

be obtained

by

joining

ny

one

of the

m

points

o

any

one

of

the

n

points.

16.

We

will

next

take the formula for the

square of

a

polynomial,

iz.

As

before,

et

0

be

the

origin

and

L

another

fixed

point

on

the circumference

of the

circle.

Take

n

other fixed

points

on

the

circle,

nd

let the

lines

joining

to

these

points

e



(

94

)

NOTE

ON

THE

TWO RELATIONS

CONNECTING

THE

DISTANCES

OF

FOUR POINTS ON

A

CIRCLE.

By

Prof.

Cayley.

Consider

a

qaadrilateral

ACD

inscribed in

a

circle;

and let

the

sides

BA^ AG^ GD^

DB and

diagonals

G and

AD be

=

c,

J,

A,

/,

a,

/

respectively

/

is

for

convenience

taken

negative,

o

that

the

equation

connecting

the

sides

and

diagonals

ay

be

A,

=a/+J^

+

cA,

=0.

We have

between

the sides

and

diagonals

nother

relation

F,

=aJc

+

a^A

+

M/+c/i/, =0,

as

is

easilyroved

geometrically;

n

fact,

recollecting

hat

the

opposite

angles

are

supplementary

o

each

other,

the

doable

area

of the

qnadrilateiul

s

=

(ic

+ gK)

edn^,

and it

is

also

=

(bh

+

cg)

sin^;

hat

is,

e

have

(he

+gd)

sin

A

(bh

+

eg)

sin

jS=

0.

But

from

the

triangles

AD and

BAG^

in

which the

angles

D^

G

are

equal

to

each

other,

e

have

c

f

c a

sini

sinj?'

sin

7

sin^

'

that

is

/sin^

+

asin-B=0;

and thence

the

required

elation

a

(Jc

+

:7*)+/(**

+

y)

^

The

distances

of the four

points

n

the

circle

are

thus

connected

by

the

two

equations

=

0,

F=0.

Considering

a,

i,

c,

f^g^h

as

the

distances

from each other of

any

four

points

n

the

plane,

e

have

between them the relation

12,

=

ay(-a'-/'

+

6*

+

i7'

c'

+

A )

+

jy(

a'+Z'^J'-Z

+

c'

+

AO

-aVc'-ayA'-J'Ay*-cy^ ,

=0;

and it is clear that this

equation

should

be

a

consequence

of

the

equations

=

0,

F=

0.

To

verifyhis,orming

the

sum

fl+

F*,

we

have

a+F'=

(a*+/ )(-ay'

+

6y

+

c'A'

+

2J(7cA)-

+

(*'+i7*)(-

*y

+

o'A*

+

ay+ 2cAa/)

+

(c*

A')(-c'A'+ay +jy

+

2a/ft 7);



96

PROP.

CAYLEY,

THE

ANHARMONIC

RATIO

EQUATION.

or,

multiplying

ut,

the

equation,

s

is

well-known,

takes

the

form

But

to

effectthe

multiplication

n the easiest

manner we

may

proceed

s

follows

:

writing

a,i,c=(a-S)(^-7),

(/3-S)(7-a),

iy-8)(a-fi),

80

that

a

+

i

+

c

=

0,

the

equation

s

The

product

f the

first

pair

of factors

is

thus the

equation

s

that

is,

and

recollecting

hat

a+

+

c

=

0,

and

writing^hc-k-ca-^-abj

r^ahc^

the

equation

ecomes

(a:+l)*-3(aj+l/a;+f3

^)(aj

l)'aj -l

0;

that

is

(^

+

aj

+

l)'+^(aj

l)'aj*

0.

But,writing

f

=

-

,

we

have

(fl*+tf+l) +^(^+l)'^=0;

or

finally

(^

+

a,

+

ly

-

^,|/;^^^?'

?(x^.

1)'=

0,

the

required

esult.



CONTENTS.

PAGB

A

new

method for the

graphical

representation

of

complex

quantities

(continued).

y

J.

Brill

81

Note

on

the

two

relations

connecting

the

distances

of four

points

on a

dide.

By

Prof.

Catlky

-

94

Note

on

the

anharmonic ratio

equation.

By

Prof. Cayley

-

- -

95

The

following

apers

have been received

:

Major

Allan

Cunningham,

On the

depression

of

differential

equations.

Mr.

B.

Tucker,

The

cosine

orthocentres

and

a

cubic

through

them.*'

Mr. Q-. Q-.

Morrice,

Note

on

the

multiplication

of

nonions.

Mr.

C.

Chree,

**

Vortices

in

a

compressible

fluid.

Articles

for insertion

will be

received

by

the

Editor,

or

by

Messrs.

Metcalfe

and

Son,

Printing

Office,

Trinity

Street,

Cambridge.



WA

No.

CXCIX.]

4

NEW

SERIES.

-rff^

[November,

887.

^^

MESSENGER

OF

MATHEMATICS.

EDITED

BY

J. W.

L.

QLA.ISUELI,

So.D.,

F.K.8.,

FELLOW OF TRINITY

COLLEGE,

CAMBRIDQE.

VOL.

XVII.

NO.

7.

MAOMILLAN

AND

00,

HonBon

aui

ffiambrfifge.

1887.

Price

One

Shilling.

MWrnAT.PR

AND

SON.

OAMBRIDOB.

_Eig']^trvv



^

u^

-^6^

NOV

15

1887

^J E

COStNE^*

ORTHOCENTRES

OF

A

TRIANGLE

AND

A

CUBIC THROUGH THEM.

By

R,

Tttcksr,

,A.

ABG

is

a

triangle,

f which

AL^

BMj

CNsre the

altitudes

co-intersecting

n

the

orthocentre

H.

LFj

LE'

are

drawn

parallel

o

AG^

AB]

MD,

MF

AB,

BG',

tod

NE.NU

BG,

GA]

the

lines

AD^

JBlS^

F

cointersect

in

d-^,

and

the lines

AD\

BE\

GF'

cointersect

in

r,.

The

equationsIn

trilinear

coordinates)

o

AD,

BE^

GF

are

respectively

J)3cos-4

a7Cos(7,

7CosB=

Jacos^,

aacos(7=c)8cos5...(i)j

and

to

AD\ BE\

GF'

are

afi

cosB=

cy

cos

A

,

by

cos

(7=

aa

cosJBj

ol

cos

A

=

5^

cos

G.

,

(ii),

hence

^^

c^

are

givenby

the

equations

a

)8

7

4A

c

cos

jB

a

cos

O

b

cos

A

a*

+

i*

+

c'

a

0

y

^

r

7y

=

T

=

r,

tanto

0 cos

u

c

COS

A

a cosx

1

= tan

CD

(iii).

From

(lii)

t

is

at once

seen

that

the

Lemoine

point

It)

s

the

mid-point

f the

join

f

o-^ c^

*'(iv)

If

now

Z Zj,5

71

,

m^]

w

n

are

the

projections

f

o-j,

a

On

the

altitudes,

nen

Bl^,

Um^^

An^

cointersect

in

12,

ana

0Z

Am,,

Bn^

in

a'

(v).

Again

Gl^j

m^^

Bn^\Bl^^Gm,^

An,

cointersect

respec^

lively

n

Sj

(a

cot

5,

b

cot

C,

c

cot

-4)

and

6,

(a

cot

C,Jcot-4,

cot-B)

*

(vi).

The

mid-point

rf)

f the

join

f

e^,e,

is

(vil),

':

J'

and lies

on

the

cifcum-Brocard

axis

;

this

axis

therefore

bisects

cr, r

12i2'

and

SjE,.

VOL,

XVII.

H



98

MR.

TUCKER,

THE

COSINE

ORTHOCENTRE

OF

A

We

readily

btain

the

following

esults

:

AE

=

V

cos

Ajc^

CE

=ab

cos

jB/c,

AE'

=

be

cos^/a,

CE'

=

b'

cos

0/a,

CD

=

a*

cos

(7/6,

^Z

=

ac

cosAjb,

CD'

=

ab

cosAjCj

BD'

=

a'

cos

J5/o,

AF

=

bc

cos

(7/

,

J5F

=

c'

cos

5/a,

AF'ê

cosA/bj

BF'âccosCjb.

..(viii).

If

7,Z,,

,/^

produced

eet

the side

AC

in

Aj,A,',

nd

^-B

in

A/,Ag,

then

if K

be

put

for

a*

+

**

c ,

e

obtain

cr,

Aj

2a*6

cos

(7/

ST,

crÂ,

2a*c

cos

/

Z,

whence

7,A,

o-,A,

2a7ir,

=(by

I.

x.)*

twice

the

intercept

made

on

jBC

by

the

T. R.

circle.f

Also

cTjAj'

2abc

cos^

/

JSr=

o-,Ag',

and therefore

a^h^'aji^'

s

a

parallelogram,/A^'

being

bisected

m

the

Lemoine

point

with like

results

for

the other

sides.

..(ix).

^

.

Ah/

rX'

Agam

^

=

BD'

AG

CD'

hence

Ah^.

AB

=

Ah^.

A

(7,

f.e.

A/A^'

s

an

anti-parallel

o

BC

(x).

Since

Ah;^WclK,

Ah^^2bc'IK,

therefore

A;A;

2abclK(^

2DF

=

2ED'

=

2FE

of

I)

...(xi).

Similarly

or

the

other

sides

0

jV,hlk^corresponding

o

A/A,'),

herefore

i.e,

these

lines

are

diameters

of

the

cosine-circle

.....

(xii);

the

equation

f

which

is

(/*

1

+

cos

-4

cos

j?

cos

(7),

=

2(aa-\-...+...)[abc inBsin

7

cos^

+.,,+.

..]...(xiii).

I

cite

The

Triplicate-ratio

ircle,

uar,

Jour.,

yol.

xix,,

No.

76,

as

I.

t

We

readily

obtain

-~^

=-^

*c.



100

MR.

TUCKER,

THE

COSINE

ORTHOCENTRES

OF

A

I

collect

together

ere

a

few

results

of

interest.

The

equation

o

aâ^^

is,

f

/

=

a*

-f

6*

+

c*,

aa(a'K^v')

+

bff(

)

+

cy(

)

=

0*..

.(xix),

to

aÔ

is

which

passes

through

5 /a,cVJ,

fl'/c,(fi,),

)....(xx),

and

l/(c a)a,

l/(a

+

J)6,

l/(J

c)c...

to

a-Ô

is

which

passes

through

c'/a,

a75, 57c, (m,),

*..(xxi).

(a +J')/a,

(b'-hc')lb,

c'

a')lc,

and

l/(a

+

J)a,

l/(

+

c)i,

l/(c

a)c,

The

join

of

G^Z,

aa

(J*

c')

...+...

=

0,

evidently

asses

through

the

middle

point

of the

join

of

^t

/i,

(x^Ôj

and the

equation

o

fi^f^^s

aa(a*-JV)+...4...=0

(xxiii).

Since

AFLE',

c.,

are

parallelograms,

herefore

AL,

FE\

c.,

mutually

bisect

each other.

Since

Cj^

a

cot

C

tano),

Bj^

=

c

cosec5

tan

w,

we

get

the

perpendicular

rom

1

on

-4Z

=

tanft)(ccosecJBsina)),

.e.

X2

is

Brocard-point

f

one

cosine

triangle,

nd

SI'

of

the

other

triangle

(xxiv).

The

lines

tjY,aj^

intersectin

Jccos-4,

a

cos

(7,

a'cosjB,

(TTj),

which

is

evidently

n

AL

(xxv).

We

note

the

following

oints

n

the

figure,

hich will

be

of

use

in

the

sequel.

[AO^

jBcTj,

n

meet

in

7rj(ccosJB,

cos^,

icos^)

'

TT,

(c

cosjB,

cos

(7,

cos

jB)

TTg

(b

cos

(7,

cos

(7,

cos

-4)

7r/(6

os

(7,

cos^,

b

cos^)

7r,'(c

os

5,

c

cos

-4,

a

cosB)

7r^(b

os

(7,

cos

(7,

cos

5)

^

(xxvi),

It is

readily

provedby

rotating

he

figures

hat

OM

is

perpendicular

o

r, r,.



TRIANGLE

AND

A

CUBIC

THROUGH THEM.

101

.

(xxviii).

IB r^j

T^

n

Tj(tan-4/a,

an

(7/

,

tan5/c)

C 7j,

4(7,

n

T^(tSLnCla^

an^B/J,

an-4/c)

-4 7

Aj^

in

T,(tanjB/a,

an

-4/

6,

tan

C/c)

ICcTj,

BcTj

n

T'j(cot-4/co8^,otC/cosjB,

ot

5/

cos

C)

4 rj,

C r,

in

T\(cot

Gj

cos

A

j

cot

B/

cos

5,

cot

-4/

os

C)

Ba-^y

a^

in

T',(cotB/cos-4,

ot

-4/

os

JB,

cot

Cj

cos

C)

( vii),

I-4 r

(?,

in

v^^cotAja,

ot

C7/6,

ot

A

jc)

Ba^j

COy

in

i;,(cotjB/a,

otjB/6,

otAjc)

Ca^^

AGj

in

v^(cotja^ cotC/6,

cot

Cjc)

iAa^j

Gj

in

i;/(cot-4/a,

ot

-4/

6,

cotJB/c)

jB r

G^,

in

i;,'(cot

/a,

cot

5/

J,

cot

Bjc)

Ca^,

BG,

in

i;;(cot/a,

cot^/J,

cot

C/c)

If

-4,5j(7

s the

firstBrocard

triangle,

hen

-4^^,

jBB,,

CC,

bisect

iî^',

^/ ',

J5'

respectively

and

intersect

as

is

well-known

on

Kiepert's

yperbola)

(xxix).

The

pole

of o-ô-,ith

regard

to

the circum-circle is

a'[ c-a*cos(jB

C)],

...,

...,

'

and

therefore lies

on

the well-known

line

....

(xxx),

5ca

+

ca^

+

a

Jy

=

0

and

of

e^6

lies

on

a

(6

c*)/a

...+...

=0,

i.e.

on

the

radical

axis

of

the

circum-

and

T.R

circles

(xxxi).

Assume

then

whence

and

therefore

and

CO

=

tan

^

tan

tan

sin

(B

î)

_

i

cos-4

sin ^,

acosJS*

cot^j

tanjB+

cotjB-

cot

G,

cot

^1

=

tan

(7

+

cot

C7-

cot

B

;

cot

(f ^

cot

^j'

tanB

+

tan

(7,

cot

9^

+

cot

9j

= taniJ

+

tan

O,

^

t

^1

+

cot^,

cot^3=

tan-4+tanjB+

tan

C

.

A

tanBtan 6'=

cot

(j \

cot

(j \

cot

^'

J

.(xxxii).



102

MR.

TUCKER,

THE

COSINE

ORTHOCENTRES

OF

A

If

c7,5C=Vr^,cr^C5=V'\,

then

8in(g^'ifr^)

^

acoag

sill

^j

i

cos

C

'

whence

cot

-^j

=

2 cot

5

+

cot'JB

tan

(7,

]

and

cot

^/

=

2

cot

(7

+

cot'C

tan

5,

[...

(xxxiii).

therefore

2

cot

^

.

cot

^'

=

4

cot'o)

3.

'

From

(viii)

e

see

that

AE,AM=:b^

cos'

A

==JN\

therefore

circle

round

-ft/jîlf

ouches

AB

at

N]

similar

results

hold

good

for

the

other

sides

(xxxiv).

The

equations

o

the

circles

LE'M^

LFN

are

respectively

a?'

+

y*

=

cy

cos

Cj

sin

vl,

and

a:'

+

y*=5iy

C08i5/3in4...

(xxxv),

where

CL,

LA

are

the

axes

;

whence,

if

D,

U

are

their

diameters,

e

have

D

+

D'

=

B

cos

(B

-

C).

The

trilinear

equations

o

the

circles

in

(xxxiv)

re

FLNj

aC=Jj[b

cos

Ccosu4a

+

o

co8 5/8

+

6

cos'CV]]

DML^

hC=L\c

cosMa

-f

c

cos

-4

cos

B^

+

a

cos*

6V]

[

ENM^ cC=L[b

cosMa

+

a

cos'Bff

+

a

couB

cos

Cy]

and

E'MLy

a

(7

=

Z/

[c

cos

^

cos

-4a

+

c

cos'^/3

6

cos'*

C7]

(xxxvi),

J? -î/,

(7

=

i

[c

cosMa

+

a

cos

C

cos

^/3

+

a

cos*

C7J,

D'LN^ cC=^L\h

cosÂa

+

a

cos*jBi8

f

i

cos

4

cos

C7],

if

(7=

0^87

+

^qt

4-

ca/8

and

i

=

aa

+

JyS+

C7,

From

(xxxv)

we

see

that

if

/ ,,

^,

p, ;

p'^,

'

p\

are

the

two sets

of

radii,

hen

P

\PtPz

=

^*^ ^^*

-^

^^*

^

^^*

7/

=

PiP\p\^

\

ap^

+

1/)^

c/ jj

2jB*

8in4

sin^

sin

C=âp\

4

Jp't

^pVJ

,.., , ,

(xxxvii).

Some

general

properties

f

the

cubics

given

by

the

equation

C;E(a +

J/9

+

C7)

(-+

A

+

-)

^k'

''

\a%

bti

cy)

we given

in

the

Messengerof

Mathematics

(vol,

l.,

1864,



TKIANdLE

AND

A

CUBIC

THROUGH

THEM,

103

p.

116)

and

in the

Reprint

from

the

Educational

Times^

(vol.

.,

p.

38)

;

we

propose

here

to

consider the cubic

Cj

=

cot

o)

tan

A

tan

B

tan

C==^k\..

(xxxviii),

which

passes

through

the

points

of

reference,hrough

0-^,cr,,

and

through

the

orthocentre.

The

equation

an

be

put

into

the

form

(aa

+

J/8

+

C7)

(hc^y

+

ca'â.

+

aJa^S)

A;'aJca^7...(xxxix),

from

whence

it is

seen

that

it touches

the

minimum

circum-

ellipse

87/a

7 /c

+

a)8/c=0

at

the

points

f

reference.

The

centroid

of

the

triangle

s the

centre

of this

ellipse,

and

HA^

HB^

HC^

which

are

normals

to

the

ellipse,

re

also

normals

to

the

cubic,

and,

from

above

statement,

are

drawn

from

a

point

on

the

cubic.

The

six

points

^

t^,

t

t,

(xxvii),

nd

T,

tan^/a,

tan

0/6,

ttÂJc (xl),*

T

,

tan

CJ

a

J

tan

-4

/J,

tan

jB/

,

are

manifestly

n

(7

and

they

also

lie

on

the

ellipse

abal3

+

bcl3y

cay

a

=

cot

cot

-4

cot

5

cot

0

=

(aa+J/3

+

(yy) (xli),

wliich

IS

concentric

and

similarlylaced

with

the

above

minimum

ellipse.

The

cubic

also

passes

through

the

six

points

^^,

cot^/a,

cotjB/J,cotC/c,

j8,,

cot

-4

/a, cotC/6, cotJB/c,

/Sj,

cot/?/

,

cot

A

I

by

cot(7/o,

;9^,

cot

5/

a,

cotC7/6,

cot-4/c,

ySg,

cotO/a,

cot

-4/

6,

cotjB/c,

)8g,

cotO/a, cotJ5/6,

cot^/c.

^

which

lie

also

on

the

ellipse

a6ai8

+

645^87

^7a

=

tan*

(aa

+

i/8

+

7)*

concentric

and

similarlylaced

with

the

minimum

ellipse*

March,

1887

.(xUi),

*

These

pc^nts

are

easily

oonstrncted

in

a

difEerent

way

from

that indicated

in

(xxTii)

for

o-,

(a,),

-

(oj),

(o),

t'

(a')

we

have

the relation

between

coor-inates

to

be

aâui

=

a*

{aa^

=

^^Pi

=

'



(

104

)

NOTE

ON

THE

MULTIPLICATION

OF

NONIONS,

By

G. O.

Morrie^.

The

object

of

the

present

note

is

to

present

the

multi-^

plication

able

of nonions in its

proper

form.

We

have

m

and

n

for

our

two

fundamental

ternary

matrices

with the

condition

nm

=

pmny

where

p

is

a

primitive

ube

root

of

unity,

i':=n'=l.

Now

in

any

? uch

multiplication

or

a

group

of

operators,

and in

particular

or

substitutions

(vide

Dyck's

Gruppen^

theorettsche

Studien

//.),

it is

important

to

consider

the

per-,

mutations of

the

elements

of

the

initial

ow

which

leads

to

any

one

of the

following

ows.

In the

present

ase

we

have

cle?^rly

non-primitive

roup,

the

substitutions

which

inter-,

change

the

triads

(1,

wi,

? *),

( ,

wn,

m*n)j

(n*,

wn*,

V)

being

cyclic,

nd also

the

substitutions

which

interchange

the letters

within the

cycles^

We

may

callthe

substitutions^

1

,

5

,

5

,

's%

8',

S*s,

8*s\

Let

us

form the

matrix

J/,

1

,

m

,

w*

,

n

,

mn

,

m*^

,

w*,

7WW*,

m*n*

5

8

will

effect

a

cyclicnterchange

f

columns,

S of

rows*



MR.

CHREE,

VORTICES

IN A

COMPRESSIBLE

FLUID.

105

Moreover,

he

simplest

orm

of

m

is

m'=

(1,

0,

0

)

,

0

.

Pt

0,

0,

P*

m

(

h

0,

0,

P%

0),

0

0, 0,

p

and

it is

easy

to

verify

hat,

or

example,

the

elements of

the

second

row

are

those

of the

matrix

formed

by

multiplying

'

by

8

(M)^

the

third

row

w'*

by

s^

(M)^

and

so

on.

June

21,

1887.

VORTICES

IN

A

COMPRESSIBLE

FLUID.

By

CharUB

Chree,

M.A.y

Fellow

of

King's

College,

ambridge.

The

following

aper

contains

certain

applications

f

the

equations

f

vortex

motion in

two

dimensions

to

a

compressi-le

fluid.

The

equations

f

vortex

motion

in

an

infinite

fluid

are*

^^dP

dN_dM\

dx

dy

dz

^^^^dL^dN.

dy

dz

dx

^

'*

dP

.

dM

dL

If

dz

dx

dy

^

du

dv dw

^

.(2),

.(3)

hen

F^-^fjj^d^dy'dz'

gives

the

value

of

the function P at

a

point

(a?,

,

z)

at

a

distance

r

from the

point

{x\

y\

'),

here

6

has

the

value

^,

the

integration

xtending

through

all

space.

If

-j

denote

partial

ifferentiation,

nd

^

differentiation

following

he

fluid,

he

equation

f

continuity

s

dp dpu dpv

dpw

_

^

dt

dx

dy

dz

^

^

or

p

ot

gee

Lamb's

Maion

of

Fluids^

p.

160,

151,

for the

meaning

of

X,

Jtf,

V,. c.



108

MR.

CHREE,

VORTICES

IN A

COMPRESSIBLE

FLUID.

be treated

exactly

as

in

an

incompressible

luid

by

the

aid

of

images.

Each

image

is

to

have

the

same

cross-section

at

every

instant

as

the real

vortex,

and

the

consequent

velocity

ue

to

variation

of

density

ill

in

every

case

be

parallel

o

the

boundaryplanes.

Consider

now

a

single

straight

ortex,

parallel

o

oz^

and

let

m

be

its

strength

and

a

its

cross-section at

time

t

;

then

the

components

of the

velocity

t

a

point

in the

surrounding

fluid,

hose

coordinates

are

x

and

y

relative

to

axes

through

the

centre

of

the

vortex

fixed

in

direction,

re

given

by

my

\ diT

X

Tir*

2ir

dt r^

1

diT

y

x

V

=

5

+

,

-

Trr

2ir

at

r

.(7).

For

our

specialbject

e

require

o

determine

the

motion

of

two

such

vortices,

hose

mutual

distance is

supposed

reat

compared

to

the

diameters

of

their

cross-sections.

Let

their

strengths

e

m^^ m^,

and

at

time

t

let

their

cross-sectionsbe

7j,

(7,,

and the

coordinates

of their

centres

(a?^,

J

and

(x,,

^)

;

then,

f

r*

=

(a?,x^)*

(y,

yj ,

e

have

dt

irr'

27r

dt

r*

dt

TTT*

^

27r

dt r'

^

^

^Jyi-y )

.

1.

^

^Sli5l

dt

TIT*

27r

dt

r*

dt

TTT* 2ir dt

f^

[S).

From

these

eqaations

e

get

Thus

if

when

s=0,r.

o?

\^ ^ '

2ir

rf

'

i

. 'i

and

r,

r

we

get

r

=

c'+-( r,+

r,-, r,-, r,).

.(9).

These

equations

lso

give

d



MR.

CHREE,

VORTICES

IN

A

COMPRESSIBLE

FLUID.

109

But

if

8

denote

the

inclination

to

ox

of the

line

joining

the

centres

of

the

vortices,

,

y,

=

r

sin

e

and

x^-x^

=

r

cose,

and

the above

equation

ecomes

dt

Trr'

Thus,

if

e

=

0

when

=

0,

we

find

,^^,r

_

^

(10).

The

case

of

a

vortex before

an

infinite

plane

is

included

in

this

solution. We

have

only

to

take

^c

as

the distance

of the

vortex

from the

plane

at

the time

^

=

0,

and make

m,

=

ij,a^=a-^^

and

^a'^'=^ r^.

In

this

case

s

remains

zero.

The

stability

f the circular form

in

one

and

in

two

straight

ortices

has been

considered,

nder

the

title

of

Linked

VorticeSjy

Prof.

J.

J.

Thomson,

in his

Motion

of

Vortex

Rings. *

The main

object

of this

paper

is

to

extend

his

treatment to

a

compressible

luid. To

render

more

easy

comparison

with

Prof. Thomson's

results,

have

followed

his notation and

method

so

far

as

possible.

Consider first

a

single,pproximately

ircular,

ortex

whose

section

is the

same

for

all

values

of

2;,

and let

the

radius

of

the

cross

section

making

an

angle

6

with

a

fixed

direction be

given

at

time

t

by

jB

=

a

+

a,jC0Sw^

/8^sinn^

(11),

the last

two

terms

being

the

types

of

any

number

of

pairs

of

terms,

while

a^,

0^

are

supposed

small

compared

with

a.

Suppose

the

vorticity

nd the

density

o

be

at

any

instant

the

same

at

all

points

in the

cross

section. At

external

points

we

have

approximately,

s

for

a

vortex

of

truly

circular

cross

section,

he

functions

'27r

dt

logr.

Assume

for

the

fluid

outside the

vortex

i^=(7-^logr

{^ cosn^

+

5,sinn )f^y

...(12),

*

Part

in.,

p.

71.



MR.

CHREE,

VORTICES

IN

A

COMPRESSIBLE

FLUID.

Ill

But

from

(11)

n0(a^Binnff4^^cosnff)

(15),

where

0 isthe

angular

velocity

ound

the

axis

of the

vortex

of

the

fluid

at

its

surface.

Thus

^

1

dif

I

dP

m

^

0

=

=-

+__--=

-

4.

terms

m a

and

p.

r

dr

ir

dd

ira

*

Thus

in

(15)

we

may

take

0 =

|

.

ira

We have also

a

second

value for the

radial

velocity

iven

by

dB_ldN

dP

dt

r

de ^

dr'

Noticing

that

57

==

-57

this

last

value of

-^

can

^

2ira

dt dt

^

dt

be reduced

to

i

(a

sinnff

pn

cosn^)

-j-

,

iTQ,

at

which

must

be

identical

with

(15).

Equating

the

coefficients

of

cosn^

and

of

sinrad,

e

find

^-

-/3,( -l)^.

rf/3.

.

,.

m

I

('^^

_.=

.(n-l)-.

whence

a/

+

^^'=

constant

(17).

Also,

eliminating3^

rom

(16),

e

get

,

d*a

.

da

rfa

.

/

^

\s

/^



112

MR.

CHREE,

VORTICES

IN

A

COMPRESSIBLE

FLUID.

Multiplying

p

by

2

-^

,

and

integrating,

e

find

a (^y+wi'(n-l)'a '=

onstant

(18).

fin

If

then

a^

and

-^-f

vanish

initially

hey

always

do

so,

the

same

is

obviously

rue

of

)3^.

Suppose

initially

^

=

0,

and

so

from

(16)

^,

=

0,

while

a =oa,,

and

so

-^/=o h

^^.

i

where

a,

is

the

initial

value

of

a

;

then

from

(18)

=

oa cos

' ( -1)

-\'i

^'11

(1^).

similarly

/S^

^^a^

sin

j

(w

1)

1

\

Thus

the

form

of

the

cross-section

at

time t

is

given

by

^

=

a +

,a

cos

[nd

^m{n^

I)

T -I

(20).

The

section

thus remains

approximately

ircular,

nd

the

disturbance

in

shape

travels

round

the

cvlinder

in

the

grai-'

dually

arying

times

given

by

m

(n

1)

I

=i2tV,

where

i

is

an

integer.

If

we

suppose

^

= constant

=

^7,.

we

get

i?

=

a4.a.co,| ^-^i^=ill

The

period

of

the

ith

revolution

of

the

disturbance

is

given

by

Ti

=

-

\

e^ff-l)

1

U

fn{n-l)

,

and

so

increases

with

t

If

7

be

positive,

,e.

if

the

cros^-

section

of

the

vortex

be

increasing.

If

r

=

constant

=

Tra'*,

nd

m^coTra^

and

the

time

be

properly

chosen,

the result

(20)

is

identical

with

that

of

rrof.

Thomson

on

his

page

74,



114

MB,

CHREE,

YORTICES

IN

A

COMPRESSIBLE

FLUID.

Since

we

suppose

afc

small

we

require

to

retain

to

our

present

egree

of

approximation

nly

Kj,

a

13^^

nd

0^.

Let

iR denote

the

radial

velocity

f

a

point

on

the

second

vortex,

and 60

the

velocity

erpendicular

o

the

radius

vector,

oth

being

taken relative

to

the

centre

of

that

vortex.

Then,

from

(21),

-w

(a'^

innff

-

P\

cosnO')

(25),

where

to

the

present

degree

of

approximation

^

Try-

But

we

have

also,

utting

r'

=

J

after

diflferentiation,

r

du

dr

r

dff

ttc

^

'

^

dP.

1

d(T.

,-,

,

The

terms

in

m^

and

in

-7^

are

introduced as

it

is

the

velocity

elative

to

the

centre

of

the second

vortex

that is

being

considered.

We

thus

get,

the

terms

in

n

being

of

course

merely

typical,

=

.|^.(a inn^./S;cosn^')+^j

-^ 8in2((?'-e)...

971.

a

+

*

-^i+j

{(a^cosns

iS^sinwe)

in(^' b)

TT

C

+

(/8^

OS

we

a^

sin

we)

cos(5'

)...|

-

-,

-^

h

cos2

{ff-

e).

*

2 ^

^

?^

^^ *

^^'^^

*

^*

^ ^^^

^

^^'

^^

-

()8^

os

ne-a

sin

we)

sin(^-

e)...}

(26).



MB.

CHREE,

YOKTICBS

IN

A GOKPBESSIBLE

FLUID.

115

These

two

values

for

It

must

be

identical. The

terms

independent

f

ff

are

so

since

-j

=

27r6

-r*

and

the

identity

extends

to

the

coefficients

of

every

sine

and

cosine

of

mul-iples

of 0^

Equating

coefficients

f

cos^,

we

get,

after

reductioni

S S

^^^

^ 2b-

a,

sin2e)

g^.

J

(a,

cos2e+/3,

in2s)

but

to

our

present

degree

of

approximation

erms

of

orders

^

or

-J

-7-

are

negligible,

o

-^

=

0;

similarly

e

get

-^

=

0.

Thus

a'j,

8\

if

originally

ero

remain

so,

whether

the

fluid

be

compressible

r

not.

To the

same

degree

of

approximation

e

find,

rom

equat-ng

the

coefficients

of

cos2 '

and

of

sin

20'

in

(25j

and

(26),

rfa'

m,

^,

tn}

.

^ .

1

dfT.

^

,^^.

^_

.

a',

^co82e+J^

.

^'

8m2.=0

(28).

dt

irb

'

TTC

27r

c

ai^

^

'

It

is

scarcely

likely

that

these

equations

admit

of

a

complete

solution,

ut

general

ideas

of

the

motion

can

be

deduced.

Supposing

for

an

instant

there

were no

vorticity,

but

only

two

columns

of

fluid

of

varying

density,

e

should

have

nij

=

wij

=

8

=

0,

and

so

e?a',

_

\

h

d T^

_

ah

da

~dt

27rc''di

V

dt

^

dt

Thus

/8',

ould

be

wholly

unaflected,

hile

a'^

ould

alter

from

its

original

alue

^a

,

according

o

the

law

-/;? '*

')-

This

shews

that

a,

would

increase

or

decrease

according

as

a

were

decreasing

r

increasing.

hus,

if

both

columns

'Vere

diminishing

n

cross *3ection,

nd

so

approaching,

here

would be

a

decided

tendency

n

both

cross^'sections

to

assume

12



116

MR.

CHREE,

VORTICES

IN

A

COMPRESSIBLE

FLUID.

an

elliptical

ort

of

outline,

he

major

axes

coinciding

ith

the

line

Joining

heir

centres.

It

is

pretty

obvious,

taking

into

consideration

the

existence of

vorticity,

hat the vibrations

will become

more

important

f

the vortices

are

approaching,

and

will

not

take

place

about

a

truly

ircular form.

If

the

density

f

the

vortices

vary

very

slowly

it

is

com-aratively

easy

to

trace

the

effect

on

the

vibrations.

Suppose

and

let

m^^

=

^a'o ,

w,

=

irb'^m.

It

what follows

we

suppose

7*

and

yt

to

remain small

during

the time

considered.

Now,

when the fluid is incom-

;res8ible,

,

and

/S',

experience,

s

is

shewn

by

Prof,

'homson,*

two

forms

of

vibration,

he

shorter

period

being

27r/ '

nd

the

longer

7r/n,

here

n={aya\+a 'b\)l ^^

(30).

If then

the

period

to

which

our

equations

re

applicable

be

supposed

to

be

at

least

several

times

greater

than

the

longerperiod

f

vibration,

e

must

have

70*^

nd

7'c'o

mall

compared

to

cDa*^

o '6'^

thus

terms

in

7

or 7'

must

be

neglected

hen

terms

in

cd

or

'

occur.

Terms

in

(7^)*

r

{y'tY

re

negligible,

nd

terms

in

n

are

small

compared

to

terms

in

a

or

a '.

Removing

the

terms

which

are

negligible

ccording

o

the above

hypothesis,

e

get

for the

equations

in

a,

and

/8',,

from

(27)

and

(28),

^'-Ko'(l-7'0i8',- a',V '(l

i70

8in2n^=ol

f'

)...(31).

?Ji

a,'(l

ry' )

',

a'^Vo'*

^

+i7'0

cos2n =0

These

give

^^

+

ft)

(1 27V)a', a\b^ioc;'ft)'

2w

-

^myt)

cos2nf,

^

+

(1

277)

/3\

a'Aa)c/(ft)'

2n

-

la Wt)

sin2nt.

*

Motion

0/

Vortex

Rings^

p.

77.



MR.

CUBEE,

YOSTICES

IN

A

COMPRESSIBLE

FLUID.

117

Suppose

a'y

0

=

)8',

hen

e

=

0,

then

to

the

same

degree

of

approximation

he

solutions

of

these

equations

re

;=^T-^^j

(l

7'0cos2n -(l

i'/0cos '

+

i

^,

1

fn^e)

sin 'fl

(32),

'^' -oT^^

(^

+

'y''^

in2ne-(l

i'/0

in 'e

+

i 7Vcos ^}

(33).

Putting

/sO

we

obtain

Prof.

Thomson's

results

on

his

page

77.

The

case

m^s- 7n

a

=

J,

which

applies

o

a

single

ortex

Earallel

o

an

infinite

wall,

has

not

been

specially

reated

7

Prof.

Thomson.

It

presents

certain

peculiarities

hich

seem

worthy

of

notice.

First

neglecting

he

compressibility,

the

equations

30)

become

^- V.- 'AV j'

0.

dt

Here

o ',

,

e

are

constants; thos,

if

oC^ O ff^^

hen

=0,

the

solutions

are

a -6V(l-cosa,'0]

,o.^

/8;=

iVsIna '

j

^^*^-

The

form of

the

cross-sectionat time

t

is thus

given

bj

5'

=

6

-

iV*

cos2^

+

6V'

cos(2^-o)'0

(35),

the wall

being

perpendicular

o

the

line

^

=

0 at

the

distance

^c.

Thus the diameter of the

vortex

perpendicular

o

the

wall suffers

a

shortening

b^c^^

nd that

parallel

o

the

wall

an

equal

lengthening,

nd

the

vortex

has

a

single

isturbance,

of

period

2irjm\

about this

altered

position.

In

the

same

case,

considering

he

compressibility

lone,

we

have

^^

=

constant,

0

say,

and

from

(29)



118

COL.

CUNNINGHAM,

DEPRESSION

OF

Thus

',

y._J^'

(36),

neglecting

igher

powers

of

(blc^Y.

The

diameter

perpendicular

o

the

wall

would thus

become

c:

'

and that

parallel

o

the wall

would

become

2(6-a',)=26-2/,

+

J^^.

Supposeoa'g=:0,

hen

the

diameter

perpendicular

o

the

wall is

obviously

he

greater

if b be

decreasing,

.e.

if the

vortex

be

approaching

he

wall

;

the

reverse

is

the

case*

if

the

vortex

be

receding

from

the

wall.

Considering

hen either the

vorticity

lone

or

the

com-ressibility

alone,

we

come

to

the

conclusion

that

a

single

vortex

in

presence

of

an

infinite

wall will

not

retain

a

truly

circular

cross-section,

nd that the deviation from

the

circular

form

varies

inversely

s

the

square

of

the

distance from

the

wall. When both

vorticity

nd

compressibility

re

considered,

the deviation

from

the

circular

form will

still

vary

inversely

as

the

square

of

the distance from

the

wall,

but the

exact

change

in the form

of

the

cross-section

could

only

be

deduced

from

a

complete

solution of

the

equations

(27)

and

(28)

for

the

case

6==0,

and

this

I

have

been

unable

to

obtain.

DEPRESSION

OF

DIFFEEENTIAL

EQUATIONS-

By

LU'CoL

Allan

Cunningham^

R,E,f

Fellow

of

King's

Coll.,

Lond.,

o.

[BeferenceB

o

Boole's

Treatise

on

Differentialquations^

nd.

Ed.]

1.

General Notation. As follows

:

a?,

y

are

always

the

variables

of

the

originalquation,

f,

u

the

variables

of

a

^'depressedequation.

(^11

ydy

( ,jyi)j

( .i

.))

(^4?J

^^e

the

variables

of

the

first,

econd,third,

nd fourth

depressed

qua-ions

formed

in

succession.



DIFFERENTIAL

EQUATIONS.

119

Lagrangian

notation

is

used

for

differential

coefficients

where

compactness

is

required

thus

:^

y\

y ,...y^''^

enote

differentiation

ith

respect

o

a?,

u\u'\...u^r)

^^

^^ ^^

^^

t^

2.

Boole's

DepressioriB.

t

is shown

in

Boole'a

Differen-ial

Equations,

hap.

X.

that

a

differential

equation

dmits

of

depression

y

one

order in

each

of the

following

ases :

i.

When

not

involving

.

ii. When

not

involving

?.

iii.

When

homogeneous

in order

1,

and

when

homogeneous

in order

fit.

iv.

When

homogeneous

in

order

oo

.

But

it

is

not shown

whether

these

depressions

an

be

either

carried

out

in

succession^

r

repeated.

3.

Object

f

Paper.

It

is

proposed

ere

to

investigate

he

possibility

f

successive

and

of

repeated

depressions,

hus

greatly

extending

he

power

of the

process

of

aepression.

As

to

successive

depression

he

following

important

esult

will

be

shown

^'

A

differential

equation

dmits

of

successive

depressions

as

follows

:

i.

By

r

orders when the

r

quantities

,

y',y ,...y^*^''

are

all

absent.

ii.

By

one

order

when

x

is

absent.

iii.

By

one

order

when

homogeneous

in

any

one

order.

iv.

By

one

order

when

homogeneous

in

any

other

order.

Thus

an

equation

ot

Involving

?,y,

y',y , . y^' *\

nd

also

homogeneous

in

two

orders,

admits

of

depression

y

(r+3)

orders.

To

prove

the above

itwill suffice

to

show

that

the

success

sive

depressions

if

applied

n

suitable

order

do

not

affect

the

remainingsingularities,

.e.

produce

depressed

quation

still

possessing

hose

singularities.



120

COL.

CUNNINGHAM,

DEPRESSION

OF

It

will also be

shown that in

certain

cases

one

of

the

depressions

an

be

repeated

ithout

affecting

ny

of

the

other

singularities;

nd

that

in

certain

cases

the

application

f the

depressionsi.,

ii.,

r

iv.

produces

an

equation

ossessing

ne

of

the

singularities

.,

ii.

not

present

in

the

original,

nd

therefore

susceptible

f

further

depression.

4,

Homogeneity,

t

is

convenient

first

to

investigate

ome

properties

f

homogeneity.

Supposing

a;,

^

to

be

quantities

f

degrees

X,

fi

respec*

tively,

hen

the

following

cale

of

degrees

obtains

:

*

Quantities...

?, y,

y',

j/\

f\

...y^

Degrees

X,

/*, /*

X,

/*

2X,

/x

3X,

../x

rX.

Also

the

degree

(N)

of

any

term

Q consisting

f

the

product

of several

of

these

quantities

s

clearly

qual

to

the

sum

of

the

degrees

f

its

component

factors

;

thus,

if

then

-AT^i^X

j/t

+

a

(/i X)

+

iS

(/A

2X)

+...p

(/A

rX)

s=2/.X

+

Jf./tt,

where,

or

shortness,

icr^

-

(a

+

2/8

+

37

+...rp),

if=j-|.(a+

iS

+

7

+..../)).

Now,

it

is obvious

from

first

principles

that

all

terms

with coefficientsof

equal

degree

connected

by

the

signs

+,

^,

=

in

any

sort

of

equation

must

necessarily

e

^

same

degree

N) throughout

he

equation

hence

the

theorem

:

Every

differential

equation

ith

coefficientsof

equal

degree

in

each

term

is

homogeneous

in

Xj

y,

y\

y ,

c.,

and

may

therefore be

depressed

ne

order.

A

homogeneous

differential

equation

being

then

a

sum

of

terms

of form

Q

of

equal

degree

N

may

be

written

s((2)=o,

and

the

homogeneity

is

expressed

y

the

following,

hich

may

be called the

equation

f

homogeneity,

L.\

+

M.fi

=

N(9,

constant

for

every

term).

Def.

The

ratio

v^fi;\

of

the

degrees

of

y

and

x

is

called the

order

of

homogeneity,

nd the

quantity

is

calledthe

degree

of

homogeneity,



122

COL.

CUNNINGHAM,

DEPRESSION OF

But

these

give,

introdacing

n

arbitrary

ultiplier

j

L

(Xj

+

*X,)

3f

(/ij

kii^

(N^

+

kN^^

also

a

constant,

and

this

result

exjresses

that the

equation

as

homogeneity

of

order

v^

=

(/i^

f

ft/*,)

(X^

+

A\))

and

therefore

of

every

order,

ince

k

is

arbitrary.

.E.D.

As

to

the

possibility

f

such

multiple

omogeneity,

he

necessary

and

suflScientonditions

are

easily

een

to

be

L

and

My

both

constant

for

every

term,

or,

writing

ut,

;?

-

(a

+

2^

+

37

+...+

r/o)

ir,

(a

constant

for

every

term)^

2+

(a+

P

+

7

+. +

p)=Jf,(a

constant

for

every

term)^

and these

conditions

can

be satisfied

by

more

than

one

distinct

system

of

values of

^, j,

a,

/S,

...p

only

when

at

least

three

of

these

quantities

nter

into

every

pair

of

systems,,

r

(which

is

the

same

thing)

when

at

least three

of

the

quantities

a,

y,

y\ ...y^^

ppear

in

every

pair

of

terms:

hence

^^

Multiple

omogeneity

s

possible

nly

in

an

equation

containing

t

least three

of

the

quantities

;,

y,

y^^

. y^^

n

every

pair

of

terms,

The

fact

that

duplex

homogeneity

nvolves

multiple

omo-eneity

bears

the

following

onsequences

:

^'

A

differential

equation

hich has

multiple

omoge-

geneity

cannot

in

general

be

depressed

ore

than

two

orders in

virtue

solely

f

homogeneity.

^'The

depression-formulse

uited

to

any

order

of

homogeneity

that is convenient

may

be

applied

to

depress

a

differential

equation

which

has

homo-eneity

of

any

two

orders.

Examples

ofmultiple

omogeneity.

t

is worth

notice

that

multiple

omogeneity

occurs* in all

final

differential

quations

of

rational

algebraic

quations

(formedby

elimination

of

all

the

constants).

6.

Depeession

i.

Absence

of

y,

y',y , ..y^'^'^

r

quan-ities).

The

depression-formulae

re

(Boole,

h.

X.,

Art.

1)

*

ThiB

property

will be

proved

in another

paper.



DIFFERENTIAL

EQUATIONS.

123

Thus

this

depression

ntroduces the

new

independent

variable

(t)only

in

place

of

the

old

one

(x)

when

present

in the

original.

As

to

the effect

on

homogeneity,

t is

clear

that

if

a?,

y

be reckoned

of

degrees

X,

/x,

then

*,

u

must

be

reckoned

like

iu,

y**^,

.e.

of

degrees

X,

/*

rX

(Art.4).

Thus

a;,

y^' \

y**\ y*^

re

replaced

y

the

quantities

,

w,

w',

...i*^ **'^f

degrees

X,

il/,ilf X),

(Jf-2X),

...{il/-

n- r)X},

writing

if=(/Lt

rX)

for

shortness

;

i.e.

by

quantities

f

like

degree

with themselves. Thus

the

degree

of

each

term

in the

de-ressed

equation

is

the

same

as

that

of the

corresponding

term

in

the

original.

Hence,

if the

original

ad

single

homogeneity

of

any

order

fi

:

X

and

degree

Nj

the

depressed

equation

has

singleomogeneity

f

order M

:

X

and

of

same

degree

N,

Similarly

f

the

original

as

multiple

omogeneity,

so

also

has

the

depressed

quation.

Thus this

depression

oes

not

destroy

he

remainingingu-arities

ii.,

iii.,

v.

;

but

it

introduces

the

new

dependent

variable

(w),

so

cannot

be

repeated

unless

*^,y *\

c.

be

also

wanting).

As

this

depression

s much

easier of

appli-ation

than

depression

i.,

ii.,v.,

it

would

usually

e

applied

fio

as

to

depressbi/

s

many

orders

as

possible

t

one

step,

and

therefore could

not

be

repeated,

ecause

y^*'^

s

hereby

supposed

present

and

would introduce

u.

7.

Depeession ii. Absence

of

X.

The

depression-formulae

are

(Boole^

h.

X.,

Art.

1),

y^t,

y'^Uy

y''

uu%

y'''uW'-^uu'%

y

=

m'w'

+

4m'm

V

+

wm ,

c.

;

and,

in

general,

^/*

V

=

(^

^^Y

**

As

th se

depression-formulae

hus

generally

ntroduce

the

new

dependent

ariable

(m),

with

one

exception

noted

here-fter

(Art.

21,

wherein

u

cancels

out)

depression

.

cannot

be

applied

fter

the

present

(except

in the

case

reserved).

Also this

depression

lways

introduces the

new

independent

variable

(t)

in

place

of

y,

and

therefore

cannot

be

repeated

when

y

was

either

originally

resent

or

when

introduced

by

depression

,

(see

remarks

at

end

of last

Article).

As

to

the

effect

on

homogeneity,

he

depression-formulae

fihew

that if

a;,

y

be

reckoned

of

degrees

X,

/a,

then

tj

u

must

be

reckoned

like

y,

y

of

degrees

fi^

fi-^X

respectively,



124

COL.

CUNNINGHAM,

DEPRESSION

OF

80

that

the

quantities

ty tt,

t*',

tt ,

t*' ,

...

t* ^

are

of

degrees

fly

(/i

X),

-

\,

X

/A,

X

2/A,

...

X

(r

l)/ .

On

referring

o

the

depression-formulaB,

t

will

be

seen

that

y,

t/\

t/'\

y^'^

re

replaced

by

homogeneous

functions

of

t,

M,

w',

...w^* *^

f

same

degrees

s

their

own.

Hence,

if

the

originalquation

have

homogeneity

of

any

order

fjb

:

X

and

of

degree

N

(so

that

each

of

its

terms

are

of

degree

-AT),

the

depressed

equation

as

homogeneity

f order

(/liX)

:

/*

(the

ratio

of

the

degrees

of

m,

f),

and

of

the

same

degree

{N}

as

the

original.

Hence

also,

if the

original

quation

aa

multiple

omogeneity,

he

depressed

ill

also

have

multiple

homogeneity.

8.

Depression

iii.

Homogeneity

of

order

v.

Here

v

may

be

zero,

but

not

infinite.

The

depression-formulae

re

(Booky

Ch.

X.,

Art.

3,

Class

II)

X:=tJuj

y^x.ty

BO

that

D^^'y

(e~/?

uDtY

{s^ ^'^/?

m

+

vt)].

The

Quantity

?

or

eJ

will

be found

to

cancel

out

of

the

depressea

equation,

hich

may

be

formed

directlyy

re-lacing

a:

by

1,

y

by

f,

y'

by {u-\-vt\

f

by

{uw'

(2i/-l)M

+

v(v-l)f},

f

by

{uV

+

ttw

+

3

(v

1)

uu'

+

(3v'

6v

+

2)

m

+

y(y-.l)(v-2)f},

c.

As these

depression-formulae

hus

generally

nvolve

both

of

the

new

variables

(ty

u)

with

certain

exceptions

onsidered

hereafter

(Arts.

1,

22)

(wherein

f,

u

cancel

out)

depressions

i.,

i.

cannot

be

applied

fter

the

present

(except

n

the

cases

reserved).

As

to

the effect

on

homogeneity,

he

depression-formulas

give



126

COL.

CUNNINGHAM,

DEPEESSION

OF

As

these

depression-formulsB

hus

generatlj

Dtrodace

tite

new

dependent

ariable

(w),

with

an

exception

noted

hereafter

(Art.

21)

(wherein

u

cancels

out),depression

.

cannot

be

applied

fter

the

present

(except

in the

ease

reserved).

Again,

as

they

introduce

the

new

independent

ariable

( )

only

in

place

of the old

one

(a;),

his

depression

oes

not

affect

the

applicability

f

depressionii.,

hich

can

therefore

be

applied

fter

the

present

if

x was

originally

bsent.

As

to

the

effect

on

homogeneity,

he

depression-formulaar

give

from

which itfollows

that,

if

x

be

reckoned

as

of

degree

f

y

and

y,

y\

y'\

c.

all

of

equal

infinite

degree

(see

Art.

4)^

then

t

must

be

reckoned

(likea?)

of

degree

I,

and

u

of

degree

1,

so

that

(by

Art.

4)

the

quantities

t^

w,

u',

w , ' ,

..

u^^

are

of

degrees

1, 1,

2,

3, 4,

...(r+1).

On

referring

o

the

depression-formulas

t will

be

seea

that

oj,

y,

y',yf\

e.

are

replacedby homogeneous

functions'

of

t w,

u\

v \

e.

of

degrees

s

below

:

Degree

of

function

-1, 0,

1,

2,

3,

.,.

r,

replacing

oj,

y,

y\

y ,

f\

...

y('*^ .

Hence

this

depression

estroys

he

homogeneity

of

a

equation

possessing

ingle

homogeneity

of

order

go

,

and

therefore

cannot

be

repeated

on

such

an

equation.

But

an

equationpossessingultiplehomogeneity

has

necessarily

homogeneity

of order

zero

(Art.5),

wherein

x

is

reckoned

of

degree

-

1,

and

y,

y',y ,

c. of

degrees

,

1,2,

c.

But

these

have

been

shewn

above

to

be

the

degrees

of

the

functions

(of

f,

tt,

u\

c.)

which

replace

hem.

Hence,

in

cases

of

original

ultiple

homogeneity,

his

depression

roduces

n

equation

ith

singh

homogeneity

f

same

degree

(N)

as

the

original

nd

of order

1

(since

f,

u

are

of

degrees

1,

1).

Hence

an

equationossessingultipleomogeneity

dmits

of

two

successive

depressions

and

no

more)

in

virtue

thereof,

viz.

Ist*

One

depressiony

formulas

for

homogeneity

of

order

go

:

thisleaves

single

homogeneity

f

order

1.

2nd* One

more

depression

y

formulse

for

homogeneity

of

order

-

1

:

this

destroys

he

homogeneity.



DIFFERENTIAL

EQUATIONS.

127

10.

Depression^FormuhB.

able

II.

shews

the

functions

t)f

,

w,

w',m ,

'

that

are

to

be

substituted

for

a:,

y,

y',

y ,

y'

in

performing

any

one

depression,

nd also the

general

sub-titution

for

y^^K

The

substitutions

for

homogeneity

re.

not

ihe

actual

values of

a?,

y,

y',

c.,

the functions

of

a:,

y

which

would

necessarily

ancel

out

of

the

depressedquation

eing

t)mitted.

11.

Elevatton-FormulcB.

Table

III.

shews

the

functions

of

a?,y,

y',

y ,

c.

that

are

to

be

substituted for

t,

u,

',m ,

c.

in

raising

he order

of

any

differential

equation

in

t^

Uj

also

the

general

substitution

for

u^K

These substitutions

are

actual

equivalencies.

Depbession-Formul^.

Table

II.

\xf

the

original

ariables

;

t,

u

the

variables in

depressed

equation].

N.B.

In

these

Tables

the

sign

H

should

be read

as

becomes^

or

may

be

changed

nto.

Depression

i.

y,

/,

y ,

...y^ '^

anting.

x

=

t^

y^'-^

,

y'^

=

',

y^'^^w ,

c.,

y(' ^)

tt .

Depression

ii.

x

absent,

y

=

t,

y'=^u,

y ^uu\

y' ==uV'+ttM ,

y'-

uV

+

4ttVtt

+

' ,

Depression

iii.

Homogeneity

f

order

y

ss

1.

ajEl,

yE^,

y^'Eu-t^ y''

uu'-Su

+

2t,

y '

=

uV

+

ttw

-

6mw'

+

llw

-

6^,

y '

ttV4

4MVw''+ttM''-

10(uV'+um'')

35wm'-50m+24^,



128

COL.

CUNNINGHAM,

DEPRESSION

OP

Depression

iii.

Homogeneityf

order

f

= O,

y '

u

V

+

uu'^ Sue/

+

2tt,

Depression

iii.

Homogeneity

f

order

v

=

I.

y'

t M' +4wVm +

tttt -

2

(mV'+

mO

-

Mtt'-f

tr,

Depression

iii.

Homogeneityf

order

y

=

2.

OJEI,

y

=

f,

y'

=

w

+

2^,

/'

=

wi*'

+

3m

+

2 ,

y'

m'm

+

wu'

+

3wu'

+

2m,

3^

=

uW+

4mVm + wm' +

2

(mV'+

ttO

-

ttw'-

2i^^

y(r)

(g-/?

^J3

J'

{s/f

m

+

20).

Depression

iii.

Homogeneityf

order

v,

xEl,

y

=

tj

y'Eu

+

vty

y

=

wM'

+

(2v-

l)w+

v(K-l)f,

y' EwV'+WM +3(v-l)t*u'+(3v'-6v+2)ti+v(v-l)(v-2)f,

+

(4v'-18v

22v-6)m

+

f(v-1)(f-2)(v-3) ,

y(-)

(e-/?

DX'

{e '^/-

+

vf)}.

Depression iv.

Homogeneityf

order

oo

.

a;=f,

y

=

l,

y'Ew,

y

=

w'-fM*,

y -

w'

+

4mm

+

3m'

+

6mV

+

u\



CONTENTS.

PAGB

Vortices

in

a compressible

fluid

(continued).

By

C.

Chreb

- -

-

113

Depression

of

differential

equations.

By

Lt.-Ck)l. Allan

Cunningham

-

118

The

following

papers

have

been

received

:

Mr.

Bnchheim,

Note

on

Matrices

in

Involution.

Prof.

Oayley,

Analytical

formulae

in

regard

to

an

octad

of

points,

A

correspondence

of

confocal Cartesians with

the

right

lines

of

a

hyperboloid,*

Note

on

the

relation

between the

distances

of

five

points

in

space.

Prof.

Mathews,

Geometry

on

a

quadric

surface.

Articles

for

insertion

will

be

received

by

the

Editor,

or

by

Messrs.

Metcalfe

and

Son,

Printing

Office,

Trinity

Street,

Cambridge.



ISVS'

jr;

7cfA

No.

CO.]

NEW SERIES.

[December,

887

THE

MESSENGER

OP

MATHEMATICS.

EDITED

BY

J.

W.

L.

GLAISHER,

So.D.,

F.R.S.,

FELLOW

OP TRINITY

COLLEGE,

CAMBRIDaE.

VOL.

XVII NO.

8.

MACMILLAN

AND

CO,

^I'iHonOon

ani

ambrilige*

1887.

Price

One

Shilling.

METCALFE

AND

SON,

CAMBEIDGB,

-AAfl'



diff iiektial

equations.

129

Elevation-Foemul^.

Table III,

[t,

the

original

ariAbles

; a;,y the

variables

of

the raised

equation].

Elevation i.

y,

y\ /',

...y^'

anting,

t^x,

u:=y^'\

'

=

y(^'\

m'=3^''%

c.,

u( =y(' ** .

Elevation

ii.

x

absent

^^.._y^y-^yyy +3y

3^ y

Elevation iii.

Homogeneityf

order

vsO

. .

aw .

' '=ft^.) ^'.

Elevation

Iii.

Homogeneity

f

order

v

1.

a;'

^

a' xy -y

'

Xxy-y

V

VOL,

XVII.



130

COL.

CUNNINQHAM,

DEPRESSION

OF

Elevation iii.

Homogeneity

f

order

v.

t

p

.

U^

p

y

U

7

Kj

oj

'

a;

^y

-^yy

\xy

-

yy

V

a

Elevation

iv.

Homogeneity

f

order

oo

,

y

y

=

-3

u(^)=Dp(yj

12.

Successive

Depression.

umming

up^

it

is

seen

that

to

apply

the four

Depressions

n

succession;

he

following

must

be attended

to

:

Depression

1.

This

should

always

be

applied

first,

s

it

does

not

affect

the

remaining

Depressions,

hereas

they

if

applied

first

will

generally

prevent

its

applicationbyintroducing ).

'Depression

i.

This

should

precede

Depression

iii.

(because

the

latter

in

general

introduces

()

;

it

changes

the order of

homogeneity (when single)

from

i/

=

/Li:\toi/

=

fi

\;/Li.

Depression

iii.

This in

general

introduces

,

and reduces

multiple

homo-eneity

to

single

of

order

v

=

1,

so

cannot

precedeDepressions

ii.

or

iv.

Hence

in

cases

of

multiple

homogeneity

the

procedure

is

;

(1)

When

not

preceded

by

Depression

iv. it

may

be

applied

once

with

any

convenient

value

of

Vy

and

again

with

the

value

v

=

1.

(2)

When

preceded

immediatelyby

Depression

iv.

it

must

be

applied

with

value

i;

=

1.

(3)

When

preceded

by Depressions

iv.

and

ii.

in

turn

it

must

be

applied

with value

v=2.

Depression

iv.

This

may

precede

either

Depressions

ii.

or

iii.,

ut

cannot

follow

Depression

iii.

(because

the

latter

leaves

homogeneity only

of

order

v

=

+

1)

;

it reduces

multiple

homogeneity

to

single

of

order

v

=

-l.



DIFFERENTIAL

EQUATIONS.

181

The

following

able

shews

the order

in

which

the

Depressions

hould

be

applied

o

equations

ontaining

wo

or

more

of

the

singularities

etailed.

Thus

in

certain

cases

there

are

two

or

more

Courses

open,

viz.

2

Couwes

in

eqnations

with

two of

the

singnlarities

os.

ii.,

ii.,

r.

8

Conises

in

equation

with

sinffolaritieB

os.

ii.,

ii

,

ir.

K2



132

COL.

CUNNINGHAM,

DEPRESSION

OP

The

particular

ourse

most

advantageous

ill

of

course

depend

on

the

nature

of

the

question

but it will

probably

generally

e

advantageous

to

use

Depression

ii.with the

general

alue of

Vy

whenever

available,

s

this

leaves

a

constant

to

be

hereafter

determined in such

away

as

to

simplify

he

integrations.

13.

Combined

Depresston-Formulm,

he

results

of

substitu-ion

by

the

several

Depressions

pplied

n

succession

may

be

readily

computed

nce

tor

all

so as

to

shew

the

result

after

any

number

of

successive

Depressions.

hese

are

exhibited

in

Tables

V,

VI,

in

which

a:,

y,

y,

y%

c.,

belong

to

the

originalquation,

^v Vv Vx'i

yd

^^'J

'

^^^^

depressed

quation,

^

J

y,

Vi^

yi\

second

3

^8^

y

\

yd

c-

y

yi

^^^^^

w

^vyA^yl^yd^^'t

fourth

By

the aid

of

such

a

Table the

first,econd,third,

r

fourth

depressed

quation

ay

be

formed

at

once

without

the

labour of

forming

the

intermediate

depressionsy

simply

substituting

he functions in the Table

belonging

o

the

required

epressedquation

or

the

original

;,

y,

y\

( c.

Successive

Depressions.

Table V.

Depression

.

Step

I

in

all

cases

.

oD

x^j y

y,j

y

y\^y

yi

y\

y\

jotu...

Two

Successive

Depressions,

Depressions

.

and

ii.,

ombined,

yi=^,i

y/=y,j

yi'=y,y2i

2^r=3^,(y,y, +y/'))

yr-y,Q/:yr^^y.y:y.''^yry

Depressions

.and iii.

(v

=

0),

combined.

x^=l,y,

=

a;,,

y,'

y

y

=

y,

(y/ 1),

3'. '2y;(y.i/. '

*:^.'y;'-. )+.y.Cy.'-)(y.'-2)(y;-

).



DIFFERENTtAL

EQUATT02(S.

135

Four Successive

Depressions.

Depressions

.,

i.,

ii.

(v

=

0),

and iii.

(y

1),

combined,

y,

E

1,

y.'

1,

y,

x

y.'

x^

(y,

+

2x,

-

1),

Depressions

.,

ii.,

ii.

(v=

1),

am? iii.

(v

=

1),

combined,

Depressions

.,

i.,

ii.

( v),

an(?

iii.

(y=i

1),

^mbined.

y,=iiy,'^i,

y,

=' ' + ',

y. '=a;,y,

(a;,

v)

(2a:,

2^

-

1),

yi

=

'.y,

(^ y/

y

+

'^r*

7

v

-

3)

+

(a'

')

.^^4

3f -

2)

(2x^

+2v-

1).

Depressions.,i.,

v.

awrf

iii.

(y

=

1),

combined.

y,

=

l,

y/=l, y.

x

y/ =y,-ir,

+

2x/,

y.

=y.

(y/

+

'^r,

S)

+

x,

(2a;,

1)

(Zx,

2).

Depressions

.,v.,

i.

anc?

iii.

(y

=

2),

combined.

y.=i,

y.'^i,y, =^.+i,

y. '='^4y.+(- 4+i)(2a;,+i),

y. '=a'/(y y;+7y.+i7a;,+6)

1.

14.

Repeated

Depression.

The several

depressions

on-idered

above

may

in

some

cases

be

repeated.

15.

Depression

.

repeated.

he substitution

=

f,

y^''^=Uj

applicable

hen

the

r

terms

y,

y',

y,

...,

y* *^

re

all

wanting,

and

which

depresses

orders

at

one

step,

is

evidently

quiva-ent

to

r

repetitions

f

this

depression,epressing

ne

order

at

each

step,

viz.

by

the

substitutions

but,

as

the

same

final result

may

also

be obtained

by

the

single

ubstitution

(x

=

tj

^^'^^=w),

here is

no

advantage

in

this

repetition.

16.

Depression

i.

repeated.

n

referring

o

the

depression-

formulae

(Art.7),

it

is

seen

that

(in

the

absence

of

x)

y,

y^



136

COL.

CUNNINaHAM,

DEPRESSION

OF

become

t,

m,

and

t

does not

recur

in

the substitutions

for

y,

y' ,

c.

Hence,

if

y

be absent from

the

original,

will

be

absent

from

the

depressedquation,

o

that

depression

l.

may

be

applied

second

time.

The

h

priori

ecognition

f the further

applicability

f

this

depression

s

more

diflScult.

Thus,

if

u

be

absent

from

the

first

depressed

quation

as

well

as

^),

this

depression

ould

be

appliedgain,

inasmuch

as

y'

=

w,

and,

as

further,

enters

into the substitutions for

y'^

and

all

higher

differential

coefficients

(Art.

7),

it is

clear

that

the

absence of

u

is

secured

only

when

y',

y'',

y,

c.

occur

in

the forms

(given

in

Table

III)

equivalent

o

u\

u^\

c.,

(and

in

no

other).

The

results

of

the double

application

f

this

depression,

applicable

hen

a?,

y

are

both

absent

from

the

original

equation,

re

shown in

Table

VII,

with

the

new

variables

of

Art.

13.

It is

probable

that

this

double

application

ill

not

often

be

advantageous,

s

the

depressiony

two

orders

might

in

the

same

case

(a?,

both

absent)

be

generally

ore

simply

efi^ected

by

the

use

of

depressions

.,

i.

applied

n

succession.

The

results of

further

application

f this

depression

cannot

be

shown

in

an

equally

compendious

way

with

the last.

As

by

Art.

7,

a

single

application

f

depression

i.

does

not

destroy

homogeneity

(merely

changing

the

order,

hen

single),

t

follows

easily

that

repeated

application

ill

not

destroyhomogeneity,merelychanging

its

order

if

single.

17.

Depression

ii.

repeated.

he

a

priori

recognition

f

repeated

pplicability

with

same

value

of

v)

is

by

no

means

simple.

The

original

quation

must

of

course

possess

homo-eneity

of

some

finite

order

(v)

in

a:, y.

Besides

which

the

depressed

quation

must

also have

homogeneity

f

the

same

finite

order

(y)

in

tyU]

this

involves that the

original

quation

should

contain

a?, y,

y',y ,

c.,

only

in

the

forms

given

in

Table

III

as

the

equivalents

f

t,

w,

u\

w ,

c.,

and

should

possess

homogeneity

f

order

v

in those functions

;

a

condition

so

complex

as

to

make the h

priori

recognition

f

the

repeated

applicability

f

this

depression

ifficult.There

is,however,

one

case

of

easy

recognition,

iz. that of

multiple

omogeneity,

in which

it

has

been

shown

(Art. 8)

that

this

depression

may

be efi'ected

twice

with

the value

v

=

1

in each

instance.

By

reasoning

similar

to

that

in

Art.

8,

it

is

seen

that

the

repeated

application

f

this

depression

enerally

estroys

the

applicability

f

depression

i.

(by

introducing

he inde-endent

variable)

also,

that

in

an

equation

possessing



DIFFERENTIAL

EQUATIONS.

137

inultipleomogeneity

s

well

as

the

property

above

mentioned,

one

application

f

this

depression

ill still

leave

multiple

homogeneity.

18.

Depression

v.

repeated.

he

h

priori

ecognition

f

repeatedapplicability

s

by

no means

simple.

The

original

equation

must

of

course

have

homogeneity

of order

oo

in

a;,

y

;

and

the

depressed

quation

ust

have the

same

in

t^

u.

This

involves

that

the

original

equation

should

involve

y,

y,

y,

c.,

only

in the forms

given

In

Table

III

as

equivalent

o

w,

u\

v \

c.,

and should further be

homogeneous

in

those functions reckoned of

equaldegree.

Ex.

/

r^

-

^]

0

is of this

kind.

As this

depression

ntroduces

the

independent

ariable

only

in

place

of

the

original

;,

if

present

in

the

originalquation,

it

IS

clear that

its

repeated

application

oes

not

affect the

applicability

f

depression

i.,

hich

can

therefore be

applied

after

repeated

pplication

f

depression

v.

if

a;

were

originally

absent.

Similarly

n

an

equationossessingultiple

omogeneity,

the

repeatedapplication

f this

depression

eaves

finally

(as

in

Art.

9)

single

homogeneity

f

order

1.

19.

RepeatedDepression

ormulae.

Table

Vll

gives

the results

of

a

double

application

f

each

of

the

depressions

i.,

ii.,

v.

Repeated DEPRESSiON-FoRMULiE.

Table

VII.

a;,

y

the

original

ariables

;

^1)

Vx

5

^j?

y%

^^

variables

of

the

first

and second

depressions,

Repeated

Depression

ii.

Absence

ofx^

y.

Repeated Depression

Iii.

Homogeneity

of

order

v \

in

aj,

y

and

in

ar^

y^.

dj=l, yEl,

y=a;,+

l;

/'

=

a;,

(y,

+

a^,

+

1),

y''= y,K(y,y/'+y -i)

+

3(y,+ar,)(y;+i)}



138

COL.

CUNNINGHAM,

DEPRESSION

OP

Repeated

Depression

iv.

Homogeneity

f

order

oo

in

Xj

y

and

in

x^^y^.

x=:x,,

y=l,

/

=

1,

y

=

y,+

l,

y ' y/'

+

3y^/

+

4y;

+

y/

+

7y/

+

6y,+

l.

20.

Other

depressible

ases.

Inasmuch

as

Depressions

ii.,

ii.,

v.

introduce

in

general

both

the

independent

nd

dependent

variables

(f,

u)

in

place

of

a?,y,

y\

c

,

it

may

happen

that

certain

functions of

a;,

y,

y\

( c.

in

the

original

give

rise

on

application

f those

depressions

o

a

depressed

equation

ree from

one or

other of the

new

variables

(m, ),

and

therefore

depressible

y

Depressions

.

or ii.,

although

the

original

quation

ay

have contained

y

or

a?,

and

was

not

therefore

depressible

y Depressions

.

or

ii.

21.

Absence

of

w,

c.

(Depression

.).

On

examining

Table

III.

it

is

at

once

seen

that

**

A

differential

equation

wanting

x,

or

homogeneous

in

any

order,

and

also

involving

,

y,

^',

,

c.

only

in

the forms

equivalent

o

,

t*'* ^,* ' ^* ,

c.

of

the

Depression

li,iii,

r

iv,

(as

the

case

may

be)

will

after

that

depres-ion be

found free from

the

r

quantities

,

t*', ,...tt^* - ,

nd

be therefore

further

depressible

y Depression

1.

22.'

Absence

of

t

(Depression

ii.).

On

examining

the

depression-formulae

f

Depression

iii.

(Table

II.),

it

is

seen

that

t enters

in

the

first

degree

only

into several of

the

substi-utions

for

y,

y',

y^\

c.,

and will

therefore

disappear

from

certain

simple

functions

thereof,

epending

n

the

value

of

Vj

as

follows

:



DIFFERENTIAL

EQUATIONS.

139

Hence

it

follows

that

''A

differential

equationhaving homogeneity

of

any

order

v

{v

not

infinite),

nd

either

not

containing

the

{y

+

1)

quantities

^

y'j

\f\

' t^^^t

or

else

containing

them

only

in the forms shewn

above

for

the

value of

v,

will after

Depression

iii.leave

an

equation

free

from

ty

and

therefore

further

depressibley Depression

ii.'*

In

the

case

of

an

equation

with

multipleomogeneity,

there

is

thus

a

large

range

of

applicabilitj,

s

the

value

of

v

maj

then

be

chosen

at

will.

23.

Preparationf

equations.

An

equation

ot

possessing

the

singularities

ere considered

may

sometimes

be

trans-ormed

into

one

possessing

hem

by

a

suitable

change

of

variables.

Removal

of

x.

The

well-known results.

When

X

e*,

xD^^D,,

f(xD:)=f(D.\

a;'.i ;

A(A-l)(A-2)...{A-(r-l)}.

When

a

+

bx

=

* ,

f[(a

+

bx)DJi^f(D.\

(aH-6^X2 ;

A(A-5)(^.-2 )...{Z),-(r-l)J},

enable

the variable

x

to

be

removed

from

equations

which

contain

x

and the

diflferentials

only

in

forms

xD^^

f(xDJ,

aT.D;^

or

only

in forms

(a+ix)Z ^,/{(a+ia:)i)J,

a-\-bxyD;.

24.

Integration,

he

use

of

these

depressions

s

chiefly

as

a

help

in

the

integration

f

high-order

on-linear

difiêren-

tial

equations.

The result of the

depressions

s either

a

diffêrential

equation

f lower

order

or

an

equation

free

of

difiêrentials

(when

the

order

of the

original

s

equal

to the

number

of its

singularities)

in the

former

case

the

depressed

difiêrential

equation

ust be

integrated.

In either

case

this

equation,

free

of

differentials,

s

the

starting

oint

for

a

series

of

ascending

steps

in

which

the

successive

depressions

re

reversed

one

by

one.

Each

reversal

of

a

depression

ives

rise

to

a

first-order

differential

equation

which

is

to

be

integrated

efore

passing

n

to

the

next

Step

:

these

correspond

o

the

depressions

or

homogeneity

and



140

COL.

CUNNINGHAM,

DEPRESSION

OP

absence

of

x.

Lastly

there will be

a

series

of

r

simple

iiitegratious,

orrespoodiug

o

the

depression

or

absence

of

if'^y^y

y

25. Conditions

for

success.

The

conditions

for the

practical

solution

of

a

high-order

ifferential

equation

by

this

process

are

1 .

The

final

depressed

equation

musl

be

either free

of

difEerentials,

r

else

a

solvible

difEei-ential

equation.

'

2 .

The first-order

differential

equations

arising

must

be

separately

Bolyible.

8 .

The final

simple

integrations(r

in

number)

must

be

separately

possible.

26.

Example.

As

an

example

of

these

principles,

he

equation

9yY-45yyV

+

4 '

=

0,

which

is

the

well known

differential

equation

f

a

conic

may

be taken.

It

wants

a:,y^

y\

and

has

multiple

omogeneity,

so

may

be

depressed

five

orders

in

all,

he result

being

an

algebraic

quation.

As shewn in

Art.

12,

there

are

three Courses open, viz.

the

Depressions

ay

be

taken

in

any

of

three

orders

as

below,

(beginning

lways

with

Depression

.

applied

o

the

utmost

extent),

The

results

are

shewn,

step

by

step,

in

Table

VIII.

:

the

final

result

(Step

IV.)

of

each

Course

might

have

been

written

down

at

once

by

using

the formulae

of

Table

V.,

without

shewing

the

intermediate

steps.

The

integration

f

the three

final

depressed

quations

f

each

Course

is

shewn

in

Table

IX.

It

has

been

thought

sufficient

to

indicate

the

leading

steps

without

shewing

the

actual

details

of the

integrations

which

would

cover

several

pages).

It will be

seen

that

in this

particular

xample

it

has

been

possible

t

each

step

to

solve the first-oider

differential

equationslgebraically^

o as

to

exhibit the differential

coeflS-

cients,

s

explicit

unctions,

pon

which

the

solution

can

be

effected

by

separation

f variables.

In

particular

n Course

l *,



142

COL.

CUNNINGHAM,

DEPRESSION OF

Course

V.

Step

IV.

Depression

ii.

(v=

1).

(the

final

depressed

quation,

herein

v

is

still

arbitrary).

Course

2*. Step

II.

Depression

i.

9^;

(M,

+

yn

-

45:r,y,y/

40y/

=

0,

(an

equationossessingultipleomogeneity).

Step

III.

Depression

v.

9^/

(y/

+

2y3 )

450^3^,

40

=

0,

^

(an

equationomogeneous

in order

v

=

-

I).

Step

IV.

Depression

ii.

y4

+

2(a?,-t)(a^,-|)

0,

(the

final

depressed

quation).

Course 3^ Step

II.

Depression

v.

9y;'-i8y,y/

4y,'=o,

(an

equationanting

or,,

and

homogeneous

in

order

v

=

1).

Step

III.

Depression

i.

(an

equationomogeneous

n order

v

=

2).

Step

IV.

Depression

ii.

(i' 2).

'r y.+2.(x,-|)(x,-i)=o,

(the

final

depressed

quation).

Differential

Equation of

Conic

Table

IX.

Integrationffinal

depressed

quationf

Course

1 ,

*

or

3*.

Course

1*.

Depressions

.,

i.,

ii.

(v

=

v\

and iii.

(v=

1).

Final

depressed

quation,

^4y4

+

2(a:,

v-|)(a.,H-v-i)

0.

Digiti

zed

by



CONTENTS.

PAGH

Depression

of

differential

equations

(continued).

By

Lt.-Gol.

Allan

Cunningham

-

-

-

-

-

- -

-

129

The

following

papers

have been received

:

Mr.

Bnchheim,

Note

on

Matrices in

Inyolution.

Prof.

Cayley, Analytical

formulae

in

regard

to

an

octad

of

points,

A

correspondence

of

confocal Cartesians

with

the

right

lines

of

a

hyperboloid/'

'^Note

on

the

relation

between

the

distances of five

points

in

space.

Prof.

Mathews,

Geometry

on

a

quadric

surface.

Mr.

L. J.

Bogers,

An

extension

of the

A.

M.

and

G-.M. theorem in

inequalities.

Articles for

insertion

will

be

received

by

the

Editor,

or

by

Messrs.

Metcalfe

and

Son,

PrintingOffice,

Trinity

Street,

Cambridge.

.



'XPn'

No.

CCIL]

NEW

SERIES.

[Februaiy,

888.

^

ir,-,...v -i.

y

tSE-

MESSENGER

OP

MATHEMATICS.

EDITED

BY

J.

W.

L.

GLAISHER, So.D.,

F.R.S.,

FBLLOW

OT

TRINITY

OOLLBGB,

CAMBRIDGB.

VOL.

XVIL

NO.

10.

MAOMILLAN

AND

00.

^l-i

ontton

anS

Cambriftfie.

1888.

Price

One

Shilling.

Digiti

MRTCALFB

Ain

BOK.

CAKBRIDQB

'm



146

MB.

BOGEKS)

AN

EXTENSION

OF

A

Firstly,

et

a,,a,, ...o^

be

integers.

Then

we

merely

have

a

particular

ase

of the well-known

theorem,

wherein

we

have

a,

quantities,

ach

equal

o

J^,

c.,

the

whole number

of them

being

a^

+

a,

+...

a^.

Secondly,

et

a,,a,,

...

be

fractional.

Let

N be the

least

common

measure

of

their

denominators

and

let

Na^^A^^ ^a,

=

-4 c.,

then

we

get

by

what

is

proved

above

\

^,

+

^,+...

J

'

*

Taking

the

real

positive

^^

root

of each

side

we

get

after

reducing

he

bracketted

fraction,

he

inequality

1).

Thirdly,

etthe a's be incommensurable.

Then

we

may

substitute for

each of

these

quantities

fractions,

hich

may

differ from

them

by

less

than

any

assigned

quantities,

nd since the

theorem

ia

true

for

the

substituted

fractions,

e

may

assume

it

also

true

for

the

given

incommensurables.

Hence

we

may

consider

(1)

as

established.

It

willbe

found

conveniently

rief

to

write

s^

for

d^+a^-^ ..j

as we

shall do henceforth.

Let

J,=

for all values of

r

from

1

to

n,

then

from

(1)

/5A*i

1

1

( '

'+

) '^ ' a .'

2),

a

well known

result.

Write

flj'

or

a,,

/

for

a,,

c c,

and

let

5i

i ''

J,

a^'^j

where

m r.

Then

(1

)

gives

Again,

let

b^

a*''

where

t

r.

Then

Q'^CO^^'^'-'-r^^

Combining

hese results

we

get

fer a)



148

MB.

ROGERS,

AN

EXTENSION

OF

A

Let

u

=

0

ia

(5),

o

that

^^f^^

8^8^J

we

see

then

that

in

the

same

way

and

so

on^

or

as we

may

better

write

it,

J^ J :I

y.

(6),

a

result

which

admits

of

easy

extension

to

n

suffixes

a,

^,

7,

... .

We

shall

now

pass

on

to

applications

f the

above

results

to

Integral

alculus.

2.

In

the

inequality

1

(3)

let

where

^j

+

wA

=

^,.

We

then

get,

after

multipyling

,

5,,

s^

by

A,

and

putting

f{ai)^yy

and

making

h

decrease

indefinitely,

{jy-dxr

wdxr'

wdxr*

o),

where

m r t^

and

the

limits

are

such

that

y**,

f^

and

y*

remain

finite

and

positive

or

allvalues

between

these

limits.

As

an

example

of

this

we

may

put

y

E

,

whence

after

changing

+

1

to

97Z,

c.,

it

follows

that

er

( )'(?)'

.

where

m r t

Here

we

take

fbr

limits

1

and

0.

From

(1)

we

may

observe

that

it

is

impossible

hat

ra

[a*

I

u'^dxy,

I

v'dx^l

(3),

where

the

limits

are

independent

f

m

and

taken

so

that

the

functions

w,

t;

should

remain

positive

etween their

re-pective

limits.

For

let

rwV^

=

^(7w).

Then,

by

(1),



CEKTAIN

THEOREM IN

INEQUALITIES.

149

But

if

(3)

were

true

we

should

have the

reverse

inequality

also

true,

which

is

impossible.

Hence

(3)

cannot

hold

good.

As

an

example

we

have

/,

0

w

+

1'

80

that

no

function

u

can

be found

such

that

/;

h

where

a,

h

are

subject

o

the afore-stated

conditions.

As

we

deduced

(1)

from

1

(3),

so

may

we

draw

from

1

(4)

that

where

m T^

and

the

limits

are

under

conditions

as

before.

If

y

=

0?,

we

get

(r+ir (7n+iy (5),

where

m r.

3.

If

we

treat

the

inequality

1

(3)

in

the

same

way

as we

deduced

1

(0

fi*^ the

well-known A.M. and

G.M.

relation,

e

shall

get

(SaJT'

(2ay) ^

(SaJT'

(1).

From

1

(5)

we

get

2a6^2a5 2ay.Say

(2),

and

from

1

(6)

^aV^^^'^'

2aJ

^af

,.,

2^^

^ 2^

26

W-

These

give

results

similar

to

2

(1),

iz.

Uyv''dxn yv'dxr- {jyv^dxT-

(4),

jyv^dx

jyv'dx

jyv'^dx

yv*dx

(5),

jyv'^^'dx

^

jyv'dx

jy^dx

^

.

Jydx

Jydx

Jydx

^

^^

where

y,

v

are

functions

of

Xy

and

the limits

are

under the

same

conditions

aa

before.



150

provided

he sines

are

all

positive.

^e

may

also

get

similar

inequalities

rom

(5)

and

(6).

As

in

2

(3),

e

may

also

show

that if

I

yv^dx

=

^

(rn\

where

o,

i,

y,

v

are

subject

o

the

same

conditions,

hen

we

cannot

have

with

conditions

as

before.

The

following

quations

re

therefore

absurd

:

I

M

dr

=

-p-

,

or

-^ r^

-.

if

w * is

always

positive

etween

a:

=

Xj

and

a:

=

x

and

ar^,

r,

are

independent

f

m.

4.

We

may

also obtain

a

few

inequalities

rom

taking

logarithms

n

1

(1),

whence

,

SaJ

SaloffJ

Fron;i

this

may

be

deduced

.

Jvydx

fvlogydx

^ ^

Jvdx

^

Jvdx

with

restrictions

as

before

as

to

limits.

These

last

inequalities

o

not

appear

to

lead

to

very

interesting

esults.

-

Ozfcod,

b9.

1,

1887.



(

151

)

GEOMETRY ON A

QUADRIC

SURFACE.

By

Prof.

Mathews.

From

a

fixed

point

P of

a

quadric

urface

project

lane

sections

of

the

surface

upon

a

plane

which is

parallel

o

the

tangent

plane

at

P;

then*

the

projections

ill

all be

similar

and

similarly

ituated.

This

is

easily

provedanalytically;

r

geometrically,

y

straining

he

stereographic

rojection

f

a

sphere-

In

particular

uppose

tnat

P is

an

umbilic

Uj

then the

projections

ecome

circles,

educing

to

straight

ines

for

sections

passing

through

17,

and

conversely

o

every

straight

line

or

circle

in the

plane

of

projectionorresponds

plane

section

of

the

surface.

To

a

coaxal

system

of

circles

corresponds

he

system

of

conies in

which

the

quadric

surface

8

is

intersected

by

an

axial

pencil

f

planes.

If the

axis

of

the

pencil

eets

8 in

two

real

points

and B

every

conic

of

the

coaxal

system

will

go

through

A and

P; if,

n

the other

hand,

A and P

are

imaginary,

no

two

conies

of

the

coaxal

system

will

intersect

in

real

points.

In

this

latter

case

there

are

two

point-

conics

of the

system,

viz.

the

points

f

contact

of

the

two

planes

f

the

pencil

hich touch

8,

The

plane

of

the

pencil

which

passes

through

U

meets

8

in

a

conic,

which

may

be

called

the

radical

axis

of

the

system.

The

axis of

the

pencil

of

planescorresponding

o

a

coaxal

system

may

be

called the

polar

axis

of

the

system.

Let

a

be the

tangent

plane

to

o

at

the

point

17,

apd

let

g

be

any

straight

ine in this

plane.

Then

the

coaxal

system

of

which

g

is

the

polar

axis

projects

nto

a

system

of

circles

having

their

radical axis

at

mfinity;

hat

is,

concentric

system.

Hence

we

get

a

construction

for

what

may

be

called

the

centroid

of

any

conic drawn

upon

8.

Namely,produce

the

plane

of

the

conic

to

meet

cd

in

a

straight

ine

g

;

then

the

point

f

contact

of the other

tangent

plane

to

8^

which

can

be

drawn

through

^,

is

the

centroid in

question.

r

again,

join

U

to

the

pole

of

the

section

;

this line

will

meet

8

in

the

required

entroid.

The

centroids

of

a

coaxal

system

lie

on

a

conic

passing

through

U,



152

PROF.

MATHEWS,

GEOMETBY

ON

A

QUADRIC

SURFACE.

Let

g

be

the

polar

xis of

a

coaxal

system

;

then,

f

g'

be

conjugate

o

g

with

regard

o

/S,

the coaxal

system

of which

g'

IS

the

polar

axis

may

be

called

conjugate

o

the

other

system.

The centroids

of each

system

he

on

the

radical

axis

of the

other.

Any

two

conies

upon

8 determine

two

points

orresponding

to

the

centres

of

similitude

of

the circles

into

which the conies

project.

They

may

be

found

as

follows.

Find

the centroids

of

the

two

conies and

through

thfem

draw

two

planes

nter-ecting

0)

(the

tangent

plane

to

8

at

TJ)

in

the

same

straight

line.

Suppose

these

planes

eet

the

given

conies

in

A^

B

and

Cy

D

respectively

then,

if the

conies

UA

(7,

UBD

meet

in

0

and

VAD^

UBC

m

0\

0,

0'

are

the

points

equired.

It

is needless

to

go

into further

detail,

or

it

isevident

that

every

known

theorem

in the

geometry

of

the

straight

ine

and

circle

gives

a

corresponding

heorem in

the

geometry

f

conies

upon

8.

For

example,

two

conies

P

and

Q

upon

8

may

be

such

that

a

finite

number

of other

conies

may

be

drawn

upon

8,

each

touching

and

Q

and

two

adjacent

conies

of the

series;

nd

this

can

be

done,

if

at

all,

n

an

infinite

number

of

ways,

c.

Or

again,

s an

example

of metrical

theorems,

et

Aj

B be

any

two

fixed

points

pon

/S,

and

U

an

umbilic.

Draw

two

conies

upon

8

passingthrough

17,

A

and

U^

B

respectively,

and

intersecting

t

a

given

angle

in

U]

then

the

locus of

their

other

point

of

intersection

is

a

conic

passingthrough

A

and

B.

Again,

to

a

conic

in the

plane

of

projection

orresponds

upon

8

a

quarticaving

a

conjugateoint

at

Z7;

and

we

have upon

8

a

kind

of

projective

eometry

by

which any

such

curve

may

be

derived

from

a

plane

section

of

8^

and

its

properties

nvestigated.

EXPKESSIONS

FOR

0(a:)

AS

A DEFINITE

INTEGRAL.

By

J. W. L.

Olaisher.

In Vol.

V.

(1876),

.

173

of

the

Messenger^

gave

without

proof

an

expression

or

0

(x)

as a

definite

integral.

This

expression

ontained several

errors

which

were

corrected

in

a

paper

in

Vol.

ill.

(1887),

p.

61-66

of the

Proceedings

f

the

i3ambridge

hilosophicalociety.

This

paper

contained

six

expressions

orthe function

0

(a:),

ith

an

explanation

f

the



(

154

)

HOMOGRAPHIO

INVARIANTS AND

QUOTIENT

DERIVATIVES.

By

A. J2.

Forayth.

1.

The

investigations

ontained in

the

present

paper

were

originally

egun

with the

purpose

of

finding

the

relation,

n

which

a

class

of

functions

called

quotient

erivatives

stand

to

reciprocants.

hese

quotient

erivatives

have,

among

other

properties,

hat

of

being

covariantive

for

homographic

rans-ormations

of the

dependent

and

the

independent

ariables,

when

applied

imultaneously

nd therefore also when

applied

separately.

But

it is

evident from

their

forms that

their

aggregate

does

not

constitute

the

complete

series of

such

functions

;

and

my

firstaim has been

to

obtain these

complete

series

for each

of

the

combinations of

homographic

trans-ormation.

The

character

of

the

invariance

is

less

general

han

that

of

M.

Halphen's

ifferential

Invariants,

hich

reproduce

hem-elves,

save

as

to

a

factor,y

the

substitutions

ax

+

by

+

c

a'x

+

b'y-^c'

a aj

+

6 y

+

c *

Functions of the kinds

herein considered

have

been

previously

suggested

y

Mr. L. J.

Rogers,*

ho

has,

except

in the

case

of

the

first

kind,

limited

his

investigations

o

the

deduction

of

the

partial

ifferential

equations

hich

are

satisfied

by

the

functions.

His aim

was

the derivation

of

homographic

reciprocants.

It

is

by

a

comparison

f

the

quotient

erivatives

with

these

homographic

reciprocants

hat

the

desired relation

has

been

obtained.

The

cubic

derivative

has been

expressed

n

terms

of

them,

but the

combination

is

not

legitimatef

or

the

preservation

f

reciprocal

nvariance.

The

relation thus

suggested

s

proved

to

be

general.

2.

Perhaps

the

simplest

ay

of

obtaining

he

quotient

derivativea

is

as

follows. Without

reproducing

he

general

investigation

n

which

they

arise,

onsider for

example

a

Homographic

and

Circular

Reciprocants

(first

aper),

roc.

Land,

MaJth.

Soe.,

vol.

XVII.

(1886),

p.

^20 231,

t

Sylvester,

Lectures

on

the

Theory

of

Bedprocants,''

mer. Joum,

MaHh,^

vol.

viii.

(1886),

.

212.



MB.

FOESTTH,

HOMOGBAPHIC

INVABIANT8.

155

cubic

equation

of

wbich

the

primitive

s

and let

y

be

the

quotient

of

two

differentsolutions

w,

and

w,

of

this

equation,

o

that

Since the

third and

every

higher

derivative

of

u

vanish,

we

have

y +3/v+

3y '

=0,

and

therefore

eliminating

^,

m^',

,

which

are

linearly

independent

of

one

another,

we

nave

(now

indicating

differentiation

by subscriptntegers)

\j/i^l^

=

0.

/sy

3y

By,

This

is

a* differential

equation

of

the

fifth

order;

its

primitive

s

_Aj^Bx+^

y'^D-\-Ex

Fx''

The

function

on

the

left-hand

side

is

called the

cubic

quotient

erivative.

The

quadratic

quation

--p^

=

0 leads

similarly

o

the

well-known

Schwarzian

the

qnartic

equation

^4

=

0

leads

to

a

similarly

ormed

quartlc

uotient

erivatire

[y ]i'

Vi,

4y. %,.

^Vi

Va

5^4*

10y

lOy,



156

MR.

FORSYTH,

HOMOORAPHIC INVARIANTS

and

60 on.

And

the

property

of

homographic

invarlance

already

eferred

to

is

constituted

by

the

equation

Some

other

properties

ill

be

obtained

later,

ne

of

them

in

particular

45)shewing

why

in

these

derivatives

the

highest

ifferential

coefficients

of

y

which

occur

are

only

those defined

by

alternate

integers.

Invariants when

the

independent

ariable

is

subject

o

hom H

graphicransformation.

3.

Let

f

y,

x)

be

a

function

which,

hen

the

independent

variable

x

is

transformed,

eproducestself,

ave

as

to

a

power

of

-T

,

BO

that

we

may

write

The

exponent

m

iscalledthe

index of

the

Invariant.

4.

Then

the

followingroperties

f the

general

forms

of

such

functions

are

easily

erived

if

(i).

The

independent

ariable

does

not

eocplicitly

ccur.

For

a:

=

-f

c,

where

c

is

an

arbitrary

onstant,

is

a

possible

transformation

;

if

x

occurs on

the

left-hand

side

of the

above

invariant

equation,

here

will,

after

the

substitution

for

a?,

arise

a

term

or

terms

in

c,

which

do

not

occur

on

the

right-

hand

side.

(ii).

he

irreducible

invariantsdo

not

explicitly

ontain

the

dependent

ariable.

For,

if

a

given

invariant

^

contain

y,

it

can

be

arranged

n

powers

of

y

in

the

form

If

the

transformed

value of

(f

e

4 ,

then

we

may

write

*

Proc.

Royal 8oe^

12th

Jan.,

1888.

t

These

oorrespond

to

the

propositionBiren

by

Halphen,

Thdse

^

Sur

Ic*

inTariantft

diff6rentielB,

.

21.



AND

QUOTIENT

DERIVATIVES..

157

and

since

^

is

an

invariant,

e

have

Henoe

*.-(|) t.+y{*,-|) *,}

....0.

Now

f ^

nd

I ^

o

not

explicitly

ontain

y,

which

is

subject

to

no

variation

;

hence

Similarly

*t-(^)

i

=

^

and

so

on;

from which

it

follows

that

^^,

^^j

^^

invariants

of

index

m.

And

they

are

all

explicitly

ree

from

the

dependent

ariable.

(iii)

nvariants

are

of

uniform

grade^

that

is,

he

sum

of

the

orders of

diflferentiation

f the

dependent

ariable in

the

factors

of

any

term

of

an

invariant

is

the

same

for all the

terms

of that

invariant.

For

a

possible

ransformation

is

z =

axj

where

a

is

a

constant

;

and

then

dx

dz^

so

that

the

effect

of

the

transformation

on

any

term

is

to

multiply

t

by

a

power

of

a

equal

to

the

grade

of

the

term.

But

the

effect

of

this

transformation

on

the

invariant

is

to

multiply

t

by

a *,

hence

the result.

From

this

it

at

once

follows

that

the

index

of

an

invariant

of

the

class

at

'present

considered

is

equal

o

its

grade,

(iv).

Irreducible

invariants

are

homogeneous

in

the

diffe-ential

coefficients

f

the

dependent

ariable.

For

let

an

invariant

^

of

index

m

be

arranged

n

the

form

where

l)p

s

the

aggregate

of

terms

in

yfr

of the

degree

p

;

then

we

have

Now

a

change

of

the

independent

variable

in

any

differential

oefficient

f

y

gives quantity

hich is

linearin



158

MB.

FOBSTTH,

HOMOQBAPHIC

INYABIANTS

the

new

differential

coefficients

;

hence

the

degree

of

a

term

in

the

invariant

is unaltered.

The last

equation

herefore

shews

that

the

aggregate

of

terms

of

any

degree

transforms

into

a

corresponding

ggregate

of

the

same

degree,

nd

that

this

aggregate

is

an

invariant.

It

IS

evident that allthese

results hold for

any

transforma-ion

of the

independent

ariable.

5.

Now

for

any

transformation

the

general

law

of

differentiation

s

where

Cr,

=

coefficient

f

p'

in

]p^i

^

p\

+

q-j

p\

+

[

J

dsi

d'z

and

z,

=

-^,z,

=

^,....

One method

of

obtaining

he

characteristic

differential

equations,

atisfied

by

the

invariants,

s

to

consider the

effect

on

the

invariant

equation

consequent

on

an

arbitrary

infinitesimal

change

in the

independent

ariable.

Such

a

change

may

be taken

in the form

where

8

is

an

infinitesimal

and

constant^

nd

/i

is

an

arbitrary

function

of

a;

;

we

then

have

and,

for

r

1,

Hence

we

deduce

Cr.r=l

+

re;A

and,

for

5

r,

n

^

and

therefore

d' d'

_

f r d-' r

.

(T*

rl

d

- ^'^^OlVld.

r

rl d^'

d

when

quantities

f

only

the

firstorder

are

retained.



AND

QUOTIENT

DERIVATIVES.

159

6.

In the

case

of

a

homographic

ransformatioD,

ay

z

=

so

that

i2?,

^05+

A'

^

eh-fg

{gx^hj

2 9

it

is

sufficient,

or

reduction

to

what

precedes,

o

take

e

=

A,

y

=

0,

(jr

^e

J

and then

^

=

1

+

so;,

so

that

fi^

=

x^

and

higher

differentialcoefficientsof

ii

vanish. The

pre-eding

general

ormula

now

becomes

d d d

^

.

^

d^'

Applying

this

to

the

equation

=

(1+W2sa;)0(y,

),

we

have

on

substituting

or

the

differential

coefficients

of

y

on

the

left-hand

side

an

equation

=

(l

+

iwsaj) ^(y,),

where

in

terms

multipliedy

s

we

may

take

or

a?

indifferently

as

the

variable.

Since this

equation

s

to

be

identically

true,

we

have

an

index-equation,

r

say

the

grade-equation

and

2r(r-l)y..g 0,

a

form-equaiion.

When

the

form

of the function

isdetermined

by

the

latter

and

the index is

inferred

from

the

form,

then

the

grade

equation

s

identically

atisfied.



160

MR.

FORSYTH,

HOMOGRAPHIC

INVARIANTS

7.

These

equations

oincide with

the

equations

therwise

obtained

by

Mr.

Rogers

(I.e.);

nd,

as

he

has

given

a

succession

of functions

satisfying

hem,

it

is

not

proposed

here

to

do

more

than

merely

to

state

results.

An

important

roposition,

hich admits

of

easy

proof

and

will

be

subsequently

sed,

is

the

following

If

^

he

an

invariant

of

grade-index

n,

then

y^

^

fny/f

is

an

invariant

ofgrade-index

+

2.

For

tM_thi$^

\dx)

\dz)

so

that

dx\y:)-^^dz\/d^^\'

or

-J-

[4^y^)

s

an

invariant

of

grade-

dex

unity,

nd

there-ore

y,

-7

wiy, ^

s

an

invariant

of

grade-index

+

2.

It

is

by

the

use

of this

proposition

hat

Mr.

Bogers

obtains

his

succession

of

educed

functions,

eginning

ith

the

customary

Schwarzian

y,y,

|y,*;

from

and

after

the second

educt,

owever,

simpler

orms

can

be

given

as

follows.

8.

The

function

of

the first

degree

is

The

functions

of

the

second

degree

re

:

f.^Vxy.

^^y.y,

losy.y.

^^y:^

and,

generally,

-

J^[

2jg

+

l 2jg )

^''^ ^r'^'^ 2^ '-'^ 22?--5-fll2

where

e,

=

i(-l)'j

and

for

*s=0,

1,

2,

...,p-

1,



CONTENTS.

PAGE

Homographic

invariants

and

quotient

derivatives

(continned). By

A.

R.

Forsyth

-

-

-

-

- - -

-

161

The

following

papers

have been

received

:

Hr.

Buchhoim,

Note

on

matrices

in involution.

Prof.

Cayley,

Analytical

formulse in

regard

to

an

octad

of

points/

A

correspondence

of

confocal Cartesians

with the

right

lines

of

a

hyperboloid/'

*'

Note

on

the

relation

between

the distances

of five

points

in

space.

Articles

for

insertion

will

be

received

by

the

Editor,

or

by

Messrs.

Metcalfe

and

Son,

Printing

Office,

Trinity

Street,

Cambridge.

The Messenger of Mathematics v17 1000034177

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Transcript of The Messenger of Mathematics v17 1000034177