| n' - a - a | = -1, ∴ n' = 2 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.
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:
| For the wheel | F, | n m | = | 20 19 | = | n' - a -a | ∴ n' = | - | 1 19 | a. | |
| " | H, | n m | = | 1 | = | n' - a -a | ∴ n' = | 0. | |||
| " | K, | n m | = | 21 20 | = | n' - a -a | ∴ n' = | + | 1 21 | a. |
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; that of F being positive and that of K negative.
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: