Substituting these expressions in the modulus, representing

4 2

by ', and observing that if U, represent the work yielded by the driven wheel during the action of each tooth, =U,, so that P,a,=,, we have

then P,a,.

2T

U1 = {A log.

2

2T

log. cos.

+log.1+tan.

tan.

Ն

tan. tan. -tan." tan.2+ &c. Substituting this

expression in the preceding equation, and neglecting terms above the first dimension in tan.

225. If the radius r of the generating circle be equal to one half the radius r, of the pitch circle of the driven wheel, according to the method generally adopted by mechanics (Art. 203.), then e=122=1=1.

2r

In this case, therefore-that is, where the flanks of the driven wheel are straight (Art. 210.)-the modulus becomes

-(1+2) sin. 24 log. cos. }

U1+NS . . (263).

226. Substituting (in equation 262.) for q′ its value

If, therefore, we assume the teeth in the driven wheel to be so numerous, or n, to be so great a number, that the third

e

power and all higher powers of tan. (-) may

2

be ne

glected as compared with its first power, and if we neglect

powers of tan. above the second,

which enter into it to be so small as to equal their respec tive tangents.

* For assume log,ɛ cos. x=a1x2+A9X*+A«x®+ ....; then differentiating,

―tan. x=2α1x+4a2x3+6a3x2+

3.5

but (Miller, Diff. Cal. p. 95.)-tan. x=-x—†æ3 – therefore, the co-efficients of these identical series, we have

2

..; equating,

Substituting these values in equation (262), and perform

1 1

2er'

U1 = { A++ (+) sin. 2 } U,+NS;

or substituting for A its value; and assuming sin. 2p= sin., since is exceedingly small,

which is the modulus of a wheel and pinion having epicy; cloidal teeth, the number of teeth n, in the driven wheel being considerable (see equation 252).

It is evident that the value of U, in the modulus (equation 261), admits of a minimum in respect to the value of e; there is, therefore, a given relation of the radius of the generating circle of the driving, to that of the driven wheel, which relation being observed in striking the epicycloidal faces and the hypocycloidal flanks of the teeth of two wheels destined to work with one another, those wheels will work with a greater economy of power than they would under any other epicycloidal forms of their teeth. This value of e may be determined by assuming the differential co-efficient of the co-efficient of U, in equation (261) equal to zero, and solving the resulting transcendental equation by the method of approximation.

2

227. THE MODULUS OF THE RACK AND PINION.

If the radius of the pitch circle of the driven wheel be supposed infinite (Art. 213.), that wheel becomes a rack, and the radius, of the driving wheel remaining of finite dimensions, the two constitute a rack and pinion. To determine the modulus of the rack and pinion in the case of teeth of any form, the number upon the pinion being great, or in the case of involute teeth and epicycloidal teeth of any number and dimensions, we have only to give to r, an infinite value in the moduli already determined in respect

to these several conditions. But it is to be observed in respect to epicycloidal teeth, that n, becomes infinite with

whilst the ratio remains finite, and retains its equality

in each of the equations (252), (254), and (261), and sub

e

stituting for in equation (262); we have

1. For the modulus of the rack and pinion when the teeth are very small, whatever may be their forms, provided that they work truly.

U1 = { 1

Lisin. 9, +sin. U,+NS . . . . . (265).

+

ar,

2. For the modulus of a rack and pinion, with involute teeth of any dimensions (see fig. 1. p. 255),

3. For the modulus of the rack and pinion, with cycloidal and epicycloidal teeth respectively (equation 261),

In each of which cases the value of N is determined by making r, infinite in equation (247).

e is infinite. The friction of the rack upon its guides is not taken into account

in the above equations.

CONICAL OR BEVIL WHEELS.

228. These wheels are used to communicate a motion of rotation to any given axis from another, inclined to the first at any angle.

Let AF be an axis to which a motion of rotation is to be communicated from another axis AE inclined to the first at any angle EAF, by means of bevil wheels.

I

D

P

Divide the angle EAF by the straight line AD, so that DO and DN, perpenN. diculars from any point D in AD upon AE and AF respectively, may be to one another as the numbers of teeth which it is required to place upon the two wheels.*

Suppose a cone to be generated by the revolution of the line AD about AE, and another by the revolution of the line AD about AF. Then if these cones were made to revolve in contact about the fixed axes AE and AF, their surfaces would roll upon one another along their whole line of contact DA, so that no part of the surface of one would slide upon that of the other, and thus the whole surface of the one cone, which passes in a given time over the line of contact AD, be equal to the whole surface of the other, which passes over that line in the same time. For it is evident that if n, times the circumference of the circle DP be equal to n, times that of the circle DI and these circles be conceived to revolve in contact carrying the cones with them, whilst the cone DAP makes n, revolutions, the cone

U

This division of the angle EAF may be made as follows:-Draw ST and UW from any points S and U in the straight lines AE and AF at right angles to those lines respectively, and having their lengths in the ratio of the numbers of teeth which it is required to place upon the two wheels; and through the extremities T and W of these lines draw TD and WD parallel to AE and AF respectively, and meeting in D. A straight line drawn from A through D will then make the required division of the angle; for if DO and DN be drawn perpendicular to AE and AF, they will evidently be equal to UW and ST, and there. fore in the required proportion of the numbers of the teeth; moreover, any other two lines drawn perpendicular to AE and AF from any

other point in AD will manifestly be in the same proportion as 70 and DN.