ing once around an equal fixed one.

Again, if the formula be general, it should apply equally well to a
train of screw wheels: let us take, for example, the single pair shown
in Fig. 8, of which, when T is fixed, the velocity ratio is unity. The
directional relation, however, depends upon the direction in which the
wheels are twisted: so that in applying the formula, we shall have
_n/m_ = +1, if the helices of both wheels are right handed, and
_n_/_m_ = -1, if they are both left handed. Thus the formula leads to
the surprising conclusion, that when A is fixed and T revolves, the
planet-wheel B will revolve about its axis twice as fast as T moves,
in one case, while in the other it will not revolve at all.

[Illustration: PLANETARY WHEEL TRAINS. Fig. 18]

A favorite illustration of the peculiarities of epicyclic mechanism,
introduced both by Prof. Willis and Prof. Goodeve, is found in the
contrivance known as Ferguson's Mechanical Paradox, shown in Fig. 18.
This consists of a fixed sun-wheel A, engaging with a planet-wheel B
of the same diameter. Upon the shaft of B are secured the three thin
wheels E, G, I, each having 20 teeth, and in gear with the three
others F, H, K, which turn freely upon a stud fixed in the train-arm,
and have respectively 19, 20, and 21 teeth. In applying the general
formula, we have the following results:

n 20 n' - a 1
For the wheel F, --- = ---- = ---------, [therefore] n' = - ---- a.
m 19 -a 19

n n' - a
" " " H, --- = 1 = --------, [therefore] n' = 0.
m -a

n 20 n' - a 1
" " " K, --- = ---- = ---------, [therefore] n' = + ---- a.
m 21 -a 21

The paradoxical appearance, then, consists in this, that although the
drivers of the three last wheels each have the same number of teeth,
yet the central one, H, having a motion of circular translation,
remains always parallel to itself, and relatively to it the upper one
seems to turn in the same direction as the train-arm, and the lower in
the contrary direction. And the appearance is accepted, too, as a
reality; being explained, agreeably to the analysis just given, by
saying that H has no absolute rotation about its axis, while the other
wheels have; t