resent the total motion of A, we have again,

m' = m + a, or m = m' - a.

This is, clearly, the motion of A relatively to the fixed frame of the
machine; and is measured from a fixed vertical line through the
center of A. Now, if we wish to express the total motion of F
relatively to the same fixed frame, we must measure it from a vertical
line through the center of F, wherever that maybe; which gives in this
case:

n' = n + a, or n = n' - a.

but with respect to the train-arm when at rest, we have:

ang. vel. A n
------------ = ---, whence again
ang. vel. F m

n' - a n
------ = --- .
m' - a m

This is the manner in which the equation is deduced by Prof. Willis,
who expressly states that it applies whether the last wheel F is or is
not concentric with the first wheel A, and also that the train may be
composed of any combinations which transmit rotation with both a
constant velocity ratio and a constant directional relation. He
designates the quantities _m'_, _n'_, _absolute revolutions_, as
distinguished from the _relative revolutions_ (that is, revolutions
relatively to the train-arm), indicated by the quantities _m_, _n_:
adding, "Hence it appears that the absolute revolutions of the wheels
of epicyclic trains are equal to the sum of their relative revolutions
to the arm, and of the arm itself, when they take place in the same
direction, and equal to the difference of these revolutions when in
the opposite direction."

In this deduction of the formula, as in that of Prof. Rankine, all the
motions are supposed to have the same direction, corresponding to that
of the hands of the clock; and in its application to any given train,
the signs of the terms must be changed in case of any contrary motion,
as explained in the preceding article.

And both the deduction and the application, in reference to these
incomplete trains in which the last wheel is carried by the
train-arm, clearly involve and depend upon the resolving of a motion
of revolution into the components of a circular translation and a
rotation, in the manner previously discussed.

[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 velocitie