e train-arm in all cases revolves around the central
axis of the system, that is, about the axis of A, and to this the axis
of F _must_ be parallel, in order to render the deduction of the
formula, as made by Prof. Willis, and also by Prof. Goodeve, correct,
or even possible.

[Illustration: PLANETARY WHEEL TRAINS. Fig. 17]

This will be seen by an examination of Fig. 17; in which A and B are
two equal spur-wheels, E and F two equal bevel wheels, B and E being
secured to the same shaft, and A being fixed to the frame H. As the
arm T goes round, B will also turn in its bearings in the same
direction: let this direction be that of the clock, when the apparatus
is viewed from above, then the motion of F will also have the same
direction, when viewed from the central vertical axis, as shown at F':
and let these directions be considered as positive. It is perfectly
clear that F will turn in its bearings, in the direction indicated, at
a rate precisely equal to that of the train-arm. Let P be a pointer
carried by F, and R a dial fixed to T; and let the pointer be vertical
when OO is the plane containing the axes of A, B, and E. Then, when F
has gone through any angle a measured from OO, the pointer will have
turned from its original vertical position through an equal angle, as
shown also at F'.

Now, there is no conceivable sense in which the motion of T can be
said to be added to the rotation of F about its axis, and the
expression "absolute revolution," as applied to the motion of the last
wheel in this train, is absolutely meaningless.

Nevertheless, Prof. Goodeve states (Elements of Mechanism, p. 165)
that "We may of course apply the general formula in the case of bevel
wheels just as in that of spur wheels." Let us try the experiment;
when the train-arm is stationary, and A released and turned to the
right, F turns to the left at the same rate, whence:

n
--- = -1; also m' = 0 when A is fixed,
m

and the equation becomes

n' - a
------ = -1, [therefore] n' = 2a:
- a

or in other words F turns _twice_ on its axis during one revolution of
T: a result too palpably absurd to require any comment. We have seen
that this identical result was obtained in the case of Fig. 15, and it
would, of course, be the same were the formula applied to Figs. 5 and
6; whereas it has never, so far as we are aware, been pretended that a
miter or a bevel wheel will make more than one rotation about its axis
in roll