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'.