Galloway, represented in Fig. 39. Upon
examination it will be seen, although we are not aware that attention has
previously been called to the fact, that this differs from the ordinary
forms of "pin gearing" only in this particular, viz., that the elementary
tooth of the driver consists of a complete branch, instead of a
comparatively small part of the hypocycloid traced by rolling the smaller
pitch-circle within the larger. It is self-evident that the hypocycloid
must return into itself at the point of beginning, without crossing: each
branch, then, must subtend an aliquot part of the circumference, and can
be traced also by another and a smaller describing circle, whose diameter
therefore must be an aliquot part of the diameter of the outer
pitch-circle; and since this last must be equal to the sum of the
diameters of the two describing circles, it follows that the radii of the
pitch circles must be to each other in the ratio of two successive
integers; and this is also the ratio of the number of pins to that of the
epicycloidal branches.

Thus in Fig. 39, the diameters of the two pitch circles are to each other
as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used. These pins
must in practice have a sensible diameter, and in order to reduce the
friction this diameter is made large, and the pins themselves are in the
form of rollers. The original hypocycloid is shown in dotted line, the
working curve being at a constant normal distance from it equal to the
radius of the roller; this forms a sort of frame or yoke, which is hung
upon cranks as in Figs. 36 and 38. The expression for the velocity ratio
is the same as in the preceding case:

V