On the Teeth o f Wheel~.

local influence, is positive: when the clouds are negative in their iowet" strata, terrestrial bodies then chan~e~ to positive~t and this. than. ge of elec, trieal condition is felt more or less according to t le constttntton of dill'trent persons, anti tile electrical intensity uf the s tar% producing complaints of heavinesss, headache, and a general listlessness, which people cannot ac- count Int. M. Peltier stated that in a storm on the 5th of May last, the inferior clouds were negative. Idem,~Igd.

.Progress o f C i v i l I ~ n g i n e e r i n g .

" On the Teeth of Wlteeb." l~g RoJJrzrta" Wir+Lis, Jaeksonian Profess, of A'atural Philosophu, in the University of Gambridge.

The geometry of the subject o[" the teeth of wheels may be considered as complete, but it appears that important additions may be made to its prac- tical applications. The general problem is, having given a tooth of any form, to determine one width shall work cor,'ectly with it. The method of effecting this may be shown in a simple, practical manner. The curve to be traced out~ which is the sl~ape of the required tootlb is the locus of the intersectiot~s of all the outlines of tile to~th in every one of its positions. The motion produced by the mere contact of the curve so traced out with the give, tu<~th, will be u,iform. This, then, is a practical mode of show- ing the lwacticability of the problem.

"l't~e e ~icycioids aml involutes have hitherto, from the facility with which they can be described been almost utfiversally employed, and practice has been cat, fined to the class uf epieycloids which work correctly with strmght lines or' circles. The defect under which such wheels labor is~ that a wheel of fifty teeth of tile same pitch will not work correctly with a wheel of one hundred teeth of the same pitch; since the diameter of the describing cir.. tie by which the epic.ycloid is formed, must be made equal to the radius of the pitch circle of the wheel with which the teeth are to work, and will therefore be twice as large in the second case as in the t!rst. Also, if the teeth be epieycloids, generated by a circle whose radius ~s equal to that of the wheel with which it is to work, which is equally correch the same re. mark applies.

Tiffs deicer was of no great consequence when the teeth were wooden~ but it is of great consequence in iron wheels, since the founder must have a new pattern of a wheel of forty teeth for every combination that it may be required to make of tiffs wheel with others. It is desirable that tl~e teeth of wheels be formed so that any tqoth may work correctly with any other of the same pitch. This is the case with involute teeth, but the obti- pqUitv of tl~e action is an t,bjection to their introduchon. The requisite

roperty may be given to epicyeloidal teeth, by employing the following pro- position. If there be two pitch circles touching each other, an epicyelandal tooth formed by causing a given describing circle to roll on the exterior ¢ir- cumferet~ce of the first, will work currectly with an interior epieycioid [armed by causing the same describing circle to roll on the interior cir- cumference of tile second.

From this, Professor Willis deduces the corollary+ that i f f o r a set o f wheels of the same pitch a constant describing circle be taken and em- ployed to trace those portions of the teeth which project beyond each pitch

12"

138 Progress o f Civil Engineering.

line by rolling on the exterior cireumferenc% and those which lie within it by rolling on its interior circumference, then any two wheels of the set will work correctly to~ether. Thi~ corollary is new~ and constitutes the basis of the system already alluded to.

I t only remains to settle the diameter of this constant describing circle. The simplest considerations serve to show that the diameter of the constant describing circle must not be greater than the radius of the pitch circle; hence, as a convenient rule, make its diameter equal to the radius of the least pitch circle of the set. This rule is perfectly general, applyin~ to racks and large wheels, as well as to annular or internal wheels. The simplicity of this above the old system is obvious, for on the old every epi- cycloid requires two circular templets ; also there must be as many tem- plets as pitch circles in the set, whereas on this system but one describin~ templet is requiled.

For machinery in which the wheels move constantly in the same direc- tion, the strength of the teeth may be nearly doubled for the same quantity of material, by disposing it so that the bucks are an involute or the arc of a circle,, tile acting faces beirut, of the usual form.

m the preceding the exact fi)rms have been described; the author then proceeds to ascertain forms sufficiently accurate for practice, and which are ares of circles. Euler suggested 'the substitution of arcs of circles of cur- vature instead of the curves themselves. The portion of a curve employed in practice is so small, that a circular arc is sufficiently accurate, provided the centre and radius with which it isstruck bedetermined by some more accu- rate methmtthan by mere trial. With this view Professor Wil l is was led to investigate a method in which the nature and properties of curves proper for teeth are entirely neglected, and a simple construction is shown by which a pair of centres may be at once assigned for a given pair of wheels, whence arcs may be struck that will answer the purpose of enabling these wheels to work correctly together.

The nature of the [notion produced by the pressure of one circular arc against another is then examined and reduced to that of a system of three rods, the middle one of which is joined to two others, movable at their other extremity about a fixed centre; and a simple construction is arrived at by which we may always find a pair of centres for which two circular ares may be struck through any point, which wilt drive each other truly lot a small distance on each side of that point. This point, when the side of a tooth consists only of a single arc, should be on the line of centres. I t is, however, more advantageous that tim tooth should consist ot" two arcs, for then there will be two points at which the action is exact--or, e a little before reaching the line of centres, the other a little after passing it.

From these investigations, the author was led to construct an instrument for setting out tl~e teeth of wt~eels, which may be used with perfect facility by the workmen, and which has been termed an Oduntograph, the application of which is fully described. The paper contains many practical observa- tions connected with this subject, tables, &c., and concludes with some di. rections for ascertaining the correct fm'm of cutters. Iast. Civ. Eng.--Lond. 3ourm

" The Canal Lifts cn the Grand Western Canal." By JA.,a~s GaviN, 2/. Inst. C. JE.

The lift, which is the subject of the following paper, was erected b Mr y . Green in the year 1835~ on the Grand Western Canal~ and has been in ope-