ugh a train of wheels, it is
desirable that that transmission should be as free from friction and as
regular as possible. This involves care in the shaping of the teeth. The
object to be aimed at is that as the wheel turns round the ratio of the
power of the driver to that of the driven wheel ("runner" or "follower")
should never vary. That is to say, whether the back part of the tooth of
the driver is acting on the tip of the tooth of the follower, or the tip
of the driver is acting on the back part of the tooth of the follower,
the leverage ratio shall always be uniform. For simplicity of
manufacture the pinion wheels are always constructed with radial leaves,
so that the surface of each tooth is a plane passing through the axis of
the wheel. The semicircular rounding of the end of the tooth is merely
ornamental. The question therefore is, suppose that it is desired by
means of a tooth on a wheel to push a plane round an axis, what is the
shape that must be given to that tooth in order that the leverage ratio
may remain unaltered?

Epicycloidal teeth.

If a curved surface, known as a "cam," press upon a plane one, both
being hinged or centred upon pivots A and B respectively (fig. 22), then
the line of action and reaction at D, the point where they touch, will
be perpendicular to their surfaces at the point of contact--that is
perpendicular to BD, and the ratio of leverage will obviously be AE:BD,
or AC:CB. Hence to cause the leverage ratio of the cam to the plane
always to remain unaltered, the cam must be so shaped that in any
position the ratio AC:CB will remain unchanged. In other words the shape
of the cam must be such that, as it moves and pushes BD before it, the
normal at the point of contact must always pass through the fixed point
C.

[Illustration: FIG. 22.--Cam and Plane.]

[Illustration: FIG. 23.]

[Illustration: FIG. 24.]

If a circle PMB roll upon another circle SPT (fig. 23) any point M on it
will generate an epicycloid MN. The radius of curvature of the curve at
M will always be MP, for the part at M is being produced by rotation
round the point P. It follows that a line from B to M will always be
tangential to the epicycloid. If the epicycloid be a cam moving as a
centre round the centre R (not shown in the figure) of the circle SPT,
the leverage it will exert upon a plane surface BM moving round a
parallel axis at B, will always be as BP to PR, that is, a constant;
whence MN is the prope