hat of F being positive and that of K negative.

[Illustration: PLANETARY WHEEL TRAINS. Fig. 18]

The Mechanical Paradox, it is clear, may be regarded as composed of
three separate trains, each of which is precisely like that of Fig.
16: and that, again, differs from the one of Fig. 15 only in the
addition of a third wheel. Now, we submit that the train shown in Fig.
17 is mechanically equivalent to that of Fig. 15; the velocity ratio
and the directional relation being the same in both. And if in Fig. 17
we remove the index P, and fix upon its shaft three wheels like E, G,
and I of Fig. 18, we shall have a combination mechanically equivalent
to Ferguson's Paradox, the three last wheels rotating in vertical
planes about horizontal axes. The relative motions of those three
wheels will be the same, obviously, as in Fig. 18; and according to
the formula their absolute motions are the same, and we are invited to
perceive that the central one does not rotate at all about its axis.

But it _does_ rotate, nevertheless; and this unquestioned fact is of
itself enough to show that there is something wrong with the formula
as applied to trains like those in question. What that something is,
we think, has been made clear by what precedes; since it is impossible
in any sense to add together motions which are unlike, it will be seen
that in order to obtain an intelligible result in cases like these,
the equation must be of the form _n'_/(_m'_ - _a_) = _n_/_m_. We shall
then have:

n 20 n' 20
For the wheel F, --- = ---- = ----, [therefore] n' = - ---- a;
m 19 -a 19

n n'
For the wheel H, --- = 1 = ----, [therefore] n' = -a;
m -a

n 20 n' 20
For the wheel K, --- = ---- = ----, [therefore] n' = - ---- a,
m 21 -a 21

which corresponds with the actual state of things; all three wheels
rotate in the same direction, the central one at the same rate as the
train arm, one a little more rapidly and the third a little more
slowly.

It is, then, absolutely necessary to make this modification in the
general formula, in order to apply it in determining the rotations of
any wheel of an epicyclic train whose axis is not parallel to that of
the sun-wheels. And in this modified form it