ing gear. The Internal pinions, _a_, _f_, are but little smaller than
the annular wheels, A, F, and are hung upon an eccentric E formed in one
solid piece with the driving shaft, D.

The action of a complete epicyclic train involves virtually and always the
action of two suns and two planets; but it has already been shown that the
two planets may merge into one piece, as in Fig. 10, where the
planet-wheel gears externally with one sun-wheel, and internally with the
other.

But the train may be reduced still further, and yet retain the essential
character of completeness in the same sense, though composed actually of
but two toothed wheels. An instance of this is shown in Fig. 36, the
annular planet being hung upon and carried by the pins of three cranks,
_c_, _c_, _c_, which are all equal and parallel to the virtual train-arm,
T. These cranks turning about fixed axes, communicate to _f_ a motion of
circular translation, which is the resultant of a revolution, _v'_, about
the axis of F in one direction, and a rotation, _v_, at the same rate in
the opposite direction about its own axis, as has been already explained.
The cranks then supply the place of a fixed sun-wheel and a planet of
equal size, with an intermediate idler for reversing the, direction of the
rotation of the planet; and the velocity of F is

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