We have here a compound train, consisting of two simple planetary trains, A - F and A' - F'; and its action is to be determined by considering them separately. First suppose T' to be removed and find the motion of F; next suppose F to be removed and T fixed, and find the rotation of F'; and finally combine these results, noting that the motion of T' is the same as that of F, and the motion of A' the same as that of T.

Then, according to the analysis of Prof. Willis, we shall have (substituting the symbol t for a in the equation of the second train, in order to avoid confusion):

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:

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 combining, we have in the latter train m' = 0, t = -a, whence

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.

PLANETARY WHEEL TRAINS. Fig. 21

In Fig. 21 is shown a combination consisting also of two distinct trains, in which, however, there is but one train-arm T turning freely upon the horizontal shaft OO, to which shaft the wheels A', F, are secured; the train-arm has two studs, upon which turn the idlers B B', and also carries the bearings of the last wheel F'; the first wheel A is annular, and fixed to the frame of the machine. Let it be required to determine the results of one revolution of the crank H, the numbers of teeth being assigned as follows:

A = 60, F = 30, A' = 60, F' = 10.

We shall then have, for the train ABF (Eq. I.),

And for the train A'B'F' (Eq. II.),

That is, the last wheel F' turns four times about the axis LL during one revolution of the crank H. But according to Profs. Willis and Goodeve, we should have for the second train:

or four and one-third revolutions of F' for one of H.

This result, no doubt, might be near enough to the truth to serve all practical purposes in the application of this mechanism to its original object, which was that of paring apples, impaled upon the fork K; but it can hardly be regarded as entirely satisfactory in a general way; nor can the analysis which renders such a result possible.