It has already been shown that the rotations of all the wheels of a planetary train, relatively to the train-arm, are the same when the arm is in motion as they would be if it were fixed. Now, in Fig. 14, let A be the first and F the last wheel of an _incomplete_ train, that is, one having but one sun-wheel. As before, let these be so connected by intermediate gearing that, when T is stationary, a rotation of A through _m_ degrees shall drive F through _n_ degrees: and also as before, let T in the same time move through _a_ degrees. Then, if _m'_ represent the total motion of A, we have again, m' = m + a, or m = m' - a. This is, clearly, the motion of A relatively to the fixed frame of the machine; and is measured from a fixed vertical line through the center of A. Now, if we wish to express the total motion of F relatively to the same fixed frame, we must measure it from a vertical line through the center of F, wherever that maybe; which gives in this case: n' = n + a, or n = n' - a. but with respect to the train-arm when at rest, we have: ang. vel. A n ------------ = ---, whence again ang. vel. F m n' - a n ------ = --- . m' - a m This is the manner in which the equation is deduced by Prof. Willis, who expressly states that it applies whether the last wheel F is or is not concentric with the first wheel A, and also that the train may be composed of any combinations which transmit rotation with both a constant velocity ratio and a constant directional relation. He designates the quantities _m'_, _n'_, _absolute revolutions_, as distinguished from the _relative revolutions_ (that is, revolutions relatively to the train-arm), indicated by the quantities _m_, _n_: adding, "Hence it appears that the absolute revolutions of the wheels of epicyclic trains are equal to the sum of their relative revolutions to the arm, and of the arm itself, when they take place in the same direction, and equal to the difference of these revolutions when in the opposite direction." In this deduction of the formula, as in that of Prof. Rankine, all the motions are supposed to have the same direction, corresponding to that of the hands of the clock; and in its application to any given train, the signs of the terms must be changed in case of any contrary motion, as explained in the preceding article. And both the deduction and the application, in reference to these incomplete trains in which the last wheel is carried by the train-arm, clearly involve and depend upon the resolving of a motion of revolution into the components of a circular translation and a rotation, in the manner previously discussed. [Illustration: PLANETARY WHEEL TRAINS. Fig. 15]