Of these results, the first is explicable as being the _absolute_ rotation of F, but the second is not; and it will be readily seen that the former would have been equally absurd, had the axis LL been inclined instead of vertical. But in either case we should find the errors neutralized upon combining the two, for according to the theory now under consideration, the wheel A', being fixed to T, turns once upon its axis each time that train arm revolves, and in the same direction; and the revolutions of T' equal the rotations of F, whence finally in train A'--F' we have: n n' - t 3. --- = 1 = --------; in which t = 0, m' = a, m m' - t n' - 0 which gives --------- = 1, or n' = a. a - 0 This is, unquestionably, correct; and indeed it is quite obvious that the effect upon F' is the same, whether we say that during a revolution of T the wheel A' turns once forward and T' not at all, or adopt the other view and assert that T' turns once backward and A' not at all. But the latter view has the advantage of giving concordant results when the trains are considered separately, and that without regard to the relative positions of the axes or the kind of gearing employed. Analyzing the action upon this hypothesis, we have: In train A--F: n n' n' --- = 1 = --------; m' = 0, [therefore] ---- = 1, or n' = -a; m m' - a -a In train A'--F': n' n' n' --- = 1 = --------; m' = 0, [therefore] ---- = 1, or n' = -t; m m' - t -t In combining, we have in the latter train m' = 0, t = -a, whence n n' n' --- = 1 = -------- gives ---- = 1, or n' = a, as before. m m' - t +a Now it happens that the only examples given by Prof. Willis of incomplete trains in which the axis of a planet-wheel whose motion is to be determined is not parallel to the central axis of the system, are similar to the one just discussed; the wheel in question being carried by a secondary train-arm which derives its motion from a wheel of the primary train. The application of his general equation in these cases gives results which agree with observed facts; and it would seem that this circumstance, in connection doubtless with the complexity of these compound trains, led him to the too hasty conclusion that the formula would hold true in all cases; although we are still left to wonder at his overlooking the fact that in these very cases the "absolute" and the "relative" rotations of the last wheel are identical. [Illustration: PLANETARY WHEEL TRAINS. Fig. 21]