arm is in motion as they would be if it were fixed. Now, in Fig. 14,
let A be the first and F the last wheel of an _incomplete_ train, that
is, one having but one sun-wheel. As before, let these be so connected
by intermediate gearing that, when T is stationary, a rotation of A
through _m_ degrees shall drive F through _n_ degrees: and also as
before, let T in the same time move through _a_ degrees. Then, if _m'_
represent 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,