Substituting Nr,r, for the factor, which it represents in equation (245), we have

Whence, performing actual division by the denominators of
the fractions in the second member of the equation, and
omitting terms of two dimensions in sin. 1, sin. 42, sin. 4 (ob-
serving that N is already of one dimension in those variables),
we have

In this expression it is assumed that the contact of the teeth
is behind the line of centres.

this case, equal to r1, and the point M is supposed to coincide with A,
L, becomes a chord of the pitch circle, and is therefore represented

by 2r, sin. DBA, or 2r, sin. (a,+3); so that

sin. a,+ sin. ẞ 2 sin.

2r, sin. (a,+3)

(a,+B) cos. § (a,+ß), 1 2r, sin. (a,+3)

= cos. (a1+ẞ).

Τ

If, therefore, we take the angle a,ß, so as to give to P, the direc

tion of a tangent at A, this expression will assume the value,

cos. 0, or-
cos. 0, or1;

219. THE MODULUS OF A SYSTEM OF TWO TOOTHed wheels.

Let n1 and n, represent the numbers of teeth in the driving and driven wheels respectively, and let it be observed that these numbers are one to another as the radii of the pitch circles of the wheels; then, multiplying both sides of we shall obtain

equation (249) by a1

Now let A represent an exceedingly small increment of the angle, through which the driven wheel is supposed to have revolved, after the point of contact P has passed the line of centres; and let it be observed that the first member

of the above equation is equal to P1a,
r2 Av
and that 2 A
* ΔΙ'
A&
ri

ΔΙ

represents the angle described by the driving wheel (Art. 204.), whilst the driven wheel describes the angle Av; whence it

follows (Art. 50.) that P ̧a, (244)

represents the work AU,

done by the driving pressure P1, whilst this angle A is described by the driven wheel,

Let now A be conceived infinitely small, so that the first member of the above equation may become the differential co-efficient of U,, in respect to 4. Let the equation, then, be integrated between the limits 0 and ; P, L1, and L, and therefore N (equation 247) being conceived to remain constant, whilst the angle is described; we shall then obtain the equation

U;=P;4;ƒ { 1 + { (-; ++!) sin. 9+ Lift sin. e; +Lzfzsin. ç2) cosec. (8+9) } d↓+N.

airi

0

where S is taken to represent the arc r2 described by the pitch circle of the driven wheel, and therefore by that of the driving wheel also, whilst the former revolves through the angle.

220. THE MODULUS OF A SYSTEM OF TWO TOOTHED wheels, THE NUMBER OF TEETH ON THE DRIVEN WHEEL BEING CONSIDERABLE, AND THE WEIGHTS OF THE WHEELS BEING TAKEN INTO ACCOUNT.

It is evident that the space traversed by the point of contact of two teeth on the face of either of them is, in this case, small as compared with the radius of its pitch circle, and that the direction of the resultant pressure R (see fig. p. 285.) upon the teeth is very nearly perpendicular to the line of centres BC, whatever may be the particular forms of the teeth; provided only that they be of such forms as will cause them to act truly with one another. In this case, therefore,

π

the angle BOR represented by + is very nearly equal to 2, and cosec. (+) = 1.

Since, moreover, RP is very nearly perpendicular to the line of centres at A, and that the point of contact P of the teeth deviates but little from that line, it is evident that the line AP represented by a differs but little from an arc of the pitch circle of the driven wheel, and that it differs the less as the supposition made at the head of this article more nearly obtains. Let us suppose to represent the angle subtended by this arc at the centre C of the pitch circle of the driven wheel, then will the arc itself be represented by r2, and therefore λ=r very nearly. Substituting this value of a in equation (250), observing that cosec. (4+4)= 1,

But the driven or working pressure P, being supposed to

remain constant, whilst any two given teeth are in action, P, represents the work U, yielded by that pressure whilst those teeth are in contact: also rb represents the space S, described by the circumference of the pitch circle of either wheel whilst this angle is described. Now let be conceived to represent the angle subtended by the pitch of one of the teeth of the driven wheel, these teeth being 2T

supposed to act only behind the line of centres, then =

n, representing the number of teeth on the driven wheel,

and (1+) = (1 + 2) == (12+ 1);

La sin. 92, +N.S... (252), }U2+N.S

which relation between the work done at the moving and working points, whilst any two given teeth are in contact, is evidently also the relation between the work similarly done, whilst any given number of teeth are in contact. It is therefore the MODULUS of any system of two toothed wheels, the numbers of whose teeth are considerable.

221. THE MODULUS OF A SYSTEM OF TWO WHEELS WITH INVOLUTE TEETH OF ANY NUMBERS AND DIMENSIONS.

The locus of the points of contact of the teeth has been shown (Art. 201.) to be in this case a straight line DE, which passes through the point of contact A of the pitch circles, and touches the circles (EF and DG) from which the involutes are struck. Let P represent any position of this point of contact, then is AP measured along the given line DE the distance represented by & in Art. 216., and the

angle CAD, which is in this case constant, is that represented

by 0. Since, moreover, the point of contact of the teeth
moves precisely as a point P upon a flexible cord DE, un-
winding from the circle EF and winding upon DG, would
(see note, p. 261.), it is evident that the distance AP, being
that which such a point would traverse whilst the pitch
circle AH revolved through a certain angle, measured
from the line of centres is precisely equal to the length of
string which would wind upon DG whilst this angle is de-
scribed by it; or to the arc of that circle which sub-
tends the angle . If, therefore, we represent the angle
ACD by, so that CD=CA cos. ACD=r2 cos. ", then
λ=r, cos. n. Substituting this value for à in equation (249),

and observing that 0+9= −n+9 = 1−(n− q), and that

cos. n sin. + La sin. + sin. } sec. (7-4)} P2+Nr.
sec.(n−ø)}P2+Nr, ... ... (253) ;

ar

...

from which equation we obtain by the same steps as in
Art. 219, observing that

is constant, ท

which is the modulus of a system of two wheels having any
given numbers of involute teeth.

222. THE INVOLUTE TOOTH OF LEAST RESISTANCE.

It is evident that the value of U, in equation (254), or of

the work which must be done upon
the driving wheel to cause a given
amount U, to be yielded by the
driven wheel is dependent for its
amount upon the value of the co-
efficient of U, in the second member
of that equation; and that this co-
efficient, again, is dependent for its
value (other things being the same)
upon the value of representing