Whether the hypothesis was made to substantiate the formula, or the formula constructed to suit the hypothesis, is not a matter of consequence. In either case, no difficulty will arise so long as the equation is applied only to cases in which, as in those here mentioned, that motion of revolution can be resolved into those components.
When the definition of an epicyclic train is restricted as it is by Prof. Rankine, the consideration of the hypothesis in question is entirely eliminated, and whether it be accepted or rejected, the whole matter is reduced to merely adding the motion of the train-arm to the rotation of each sun-wheel.
But in attempting to apply this formula in analyzing the action of an incomplete train, we are required to add this motion of the train-arm, not only to that of a sun-wheel, but to that of a planet-wheel. This is evidently possible in the examples shown in Figs. 15 and 16, because the motions to be added are in all respects similar: the trains are composed of spur-wheels, and the motions, whether of revolution, translation, or rotation, take place in parallel planes perpendicular to parallel axes. This condition, which we have emphasized, be it observed, must hold true with regard to the motions of the first and last wheels and the train-arm, in order to make this addition possible. It is not essential that spur-wheels should be used exclusively or even at all; for instance, in Fig. 16, A and F may be made bevel or screw-wheels, without affecting the action or the analysis; but the 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.
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 m | = -1; also m'= 0 when A is fixed, |
and the equation becomes