[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 rolling 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.