applies equally well to
the original arrangement of Ferguson's paradox, if we abandon the
artificial distinction between "absolute" and "relative" rotations of
the planet-wheels, and regard a spur-wheel, like any other, as
rotating on its axis when it turns in its bearings; the action of the
device shown in Fig. 18 being thus explained by saying that the wheel
H turns once backward during each forward revolution of the train-arm,
while F turns a little more and K a little less than once, in the same
direction. In this way the classification and analysis of these
combinations are made more simple and consistent, and the
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
formulae, 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

-a = a - m', or m = 2a.

The other view of the question is, that F turns once backward in its
bearings during each forward revolution of T; whence in Eq. 2 we
have--

n' n
-------- = --- = -1, n' = -a,
m' - a m

-a
[therefore] -------- -1, which gives -a = a - m', or m' = 2a,
m' - a

as before.

It is next to be remarked, that the errors which arise from applying
Eq. I. to incomplete trains may in some cases counterbalance and
neutralize each other, so that the final result is correct.

[Illustration: PLANETARY WHEEL TRAINS. Fig. 20]

For example, take the c