Now L representing the value of L when x=0, and remaining constant,

Let now the angle ADB, made in respect to the first element of the driving wheel between the perpendicular BD or a, and the chord AD or L, be represented by 1, and let represent the corresponding angle in the driven wheel, then

(H)2=(H)*—2 (La) (*) cos. n, = (-)* {1–2 (,) (-) cos. n. };

Extracting the square root of the binomial, and neglecting

Substituting these values in the modulus (equation 272),

P2 sin.

$2

r2

{

}

Now let the angle BCG, or the inclination of the axes, from one to the other of which motion is transferred by the wheels, be represented by 2; therefore +12=21. Also

Substituting in the preceding relation, between U, and U„,

which is the modulus of the conical or bevil wheel, neglecting the influence of the weight of the wheel.

If for cos., and cos. 7, we substitute their values (see p. 323), we shall obtain by reduction

from which equation it is manifest that the most favourable directions of the driving or working pressures are those determined by the equations

232. It is evident, that if the plane of the revolution of such a wheel be vertical, the influence of its weight must be very nearly the same as that of a cylindrical or spur wheel of the same weight, having a radius equal to the mean radius of the conical wheel, and revolving also in a vertical plane. If the axis of the wheel be not horizontal, its weight must be resolved into two pressures, one acting in the plane of the wheel, and the other at right angles to it; the latter is

2

effective only on the extremity of the axis, where it is borne as by a pivot, so that the work expended by reason of it may be determined by Art. 175., and will be found to present itself under the form of N,. S, where N, is a constant and S the space described by the pitch circle of the wheel, whilst the work U1 is done. The resolved weight in the plane of the wheel must be substituted for the weight of the wheel in equation (247), which determines the value of N. Assuming the value of N, this substitution being made, to be represented by N,, the whole of the second term of the modulus will thus present itself under the form (N,+N2)S.

233. Comparing the modulus of a system of two conical wheels with that of a system of two cylindrical wheels (equation 252), it will be seen that the fractional excess of the work U2 lost by the friction of the latter over that lost by the friction of the former is represented by the formula

2

The first term of this expression is due to the friction of the teeth of the wheels alone, as distinguished from the friction of their axes; the latter is due exclusively to the friction of the axes. Both terms are essentially positive, since n

π

and 2 are in every case less than 2

Thus, then, it appears that the loss of power due to the friction of bevil wheels is (other things being the same) essentially less than that due to the friction of spur wheels, so that there is an economy of power in the substitution of a bevil for a spur wheel wherever such substitution is practicable. This result is entirely consistent with the experience

of engineers, to whom it is well known that bevil wheels run lighter than spur wheels.

234. THE MODULUS OF A TRAIN OF WHEELS.

In a train of wheels such as that shown in the accompanying

figure, let the radii of their pitch circles be represented in order by 71, 72, 73.74, beginning from the driving wheel; and let a, represent the perpendicular distance of the driving pressure

from the centre of that wheel, and a that of the driven pressure or resistance from the centre of the last wheel of the train; U, the work done upon the first wheel, u, the work yielded by the second wheel to the third, u, that yielded by the fourth to the fifth, &c., and U, the work yielded by the last or nth wheel upon the resistance, then is the relation between U, and u, determined by the modulus (equation 252), it being observed that the point of application of the resistance on the second wheel is its point of contact b with the third wheel, so that in this case a=r3•

These substitutions being made, and L. being taken to represent the distance between the point b and the projection of the point a upon the third wheel, we have

To determine, in like manner, the relation between u, and u,, or the modulus of the third and fourth wheels, let it be observed that the work u, which drives the third wheel has been considered to be done upon it at its point of contact b with the fourth; so that in this case the distance between the point of contact of the driving and driven wheels and the foot of the perpendicular let fall upon the driving pressure

* See note, p. 292.