incongruities above pointed out are avoided; since, without regard to
the kind of gearing employed or the relative positions of the axes, we
have the two equations:

n' - a n
I. -------- = ---, for all complete trains;
m' - a m

n' n
II. -------- = ---, for all incomplete trains.
m' - a m

[Illustration: PLANETARY WHEEL TRAINS. Fig. 19]

As another example of the difference in the application of these
formulæ, let us take Watt's sun and planet wheels, Fig. 19. This
device, as is well known, was employed by the illustrious inventor as
a substitute for the crank, which some one had succeeded in patenting.
It consists merely of two wheels A and F connected by the link T; A
being keyed on the shaft of the engine and F being rigidly secured to
the connecting-rod. Suppose the rod to be of infinite length, so as to
remain always parallel to itself, and the two wheels to be of equal
size.

Then, according to Prof. Willis' analysis, we shall have--

n' - a n -s
-------- = --- = -1, n' = 0, [therefore] -------- = -1, whence
m' - a m m' - a