ce 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]

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.),

n 60 n' - a