s, 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 th