[Illustration:  PLANETARY WHEEL TRAINS.  Fig. 15]

To illustrate:  Take the simple case of two equal wheels, Fig. 15, of which the central one A is fixed.  Supposing first A for the moment released and the arm to be fixed, we see that the two wheels will turn in opposite directions with equal velocities, which gives n/_m_ = -1; but when A is fixed and T revolves, we have m’ = 0, whence in the general formula

n’ — a
------   = -1, or n’ = 2 a;
-a

which means, being interpreted, that F makes two rotations about its axis during one revolution of T, and in the same direction.  Again, let A and F be equal in the 3-wheel train, Fig. 16, the former being fixed as before.  In this case we have: 

   n
  —–­ = 1, m’ = 0, which gives
   m

n’ — a
------- = 1, [therefore] n’ = 0;
-a

that is to say, the wheel F, which now evidently has a motion of circular translation, does not rotate at all about its axis during the revolution of the train-arm.

[Illustration:  PLANETARY WHEEL TRAINS.  Fig. 16]

All this is perfectly consistent, clearly, with the hypothesis that the motion of circular translation is a simple one, and the motion of revolution about a fixed axis is a compound one.

Whether the hypothesis was made to substantiate the formula, or the formula constructed to suit the hypothesis, is not a matter of consequence.  In either case, no difficulty will arise so long as the equation is applied only to cases in which, as in those here mentioned, that motion of revolution can be resolved into those components.

When the definition of an epicyclic train is restricted as it is by Prof.  Rankine, the consideration of the hypothesis in question is entirely eliminated, and whether it be accepted or rejected, the whole matter is reduced to merely adding the motion of the train-arm to the rotation of each sun-wheel.

But in attempting to apply this formula in analyzing the action of an incomplete train, we are required to add this motion of the train-arm, not only to that of a sun-wheel, but to that of a planet-wheel.  This is evidently possible in the examples shown in Figs. 15 and 16, because the motions to be added are in all respects similar:  the trains are composed of spur-wheels, and the motions, whether of revolution, translation, or rotation, take place in parallel planes perpendicular to parallel axes.  This condition, which we have emphasized, be it observed, must hold true with regard to the motions of the first and last wheels and the train-arm, in order to make this addition possible.  It is not essential that spur-wheels should be used exclusively or even at all; for instance, in Fig. 16, A and F may be made bevel or screw-wheels, without affecting the action or the analysis; but the train-arm in all cases revolves around the central axis of the system, that is, about the axis of A, and to this the axis of F must be parallel, in order to render the deduction of the formula, as made by Prof.  Willis, and also by Prof.  Goodeve, correct, or even possible.