V’= v’(1 — f/F).

A modification of this train better suited for practical use is shown in Fig. 37, in which the sun-wheel, instead of the planet, is annular, and the latter is carried by the two eccentrics, E, E, whose throw is equal to the difference between the diameters of the two pitch circles; these eccentrics must, of course, be driven in the same direction and at equal speeds, like the cranks in Fig. 36.

[Illustration:  PLANETARY WHEEL TRAINS.]

A curious arrangement of pin-gearing is shown in Fig. 38:  in this case the diameter of the pinion is half that of the annular wheel, and the latter being the driver, the elementary hypocycloidal faces of its teeth are diameters of its pitch circle; the derived working tooth-outlines for pins of sensible diameter are parallels to these diameters, of which fact advantage is taken to make the pins turn in blocks which slide in straight slots as shown.  The formula is the same as that for Fig. 36, viz.: 

V’ = v’(1 — f/F),

which, since f = 2F, reduces to V’ = -v’.

Of the same general nature is the combination known as the “Epicycloidal Multiplying Gear” of Elihu 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: 

V¹ = v’(1 — f/F); which in Fig. 39 gives

V¹ = v’(1 — 5/4)= -1/4v’: