As another example of the difference in the application of these formulae, let us take Watt's sun and planet wheels, Fig. 19. This device, as is well known, was employed by the illustrious inventor as a substitute for the crank, which some one had succeeded in patenting. It consists merely of two wheels A and F connected by the link T; A being keyed on the shaft of the engine and F being rigidly secured to the connecting-rod. Suppose the rod to be of infinite length, so as to remain always parallel to itself, and the two wheels to be of equal size. Then, according to Prof. Willis' analysis, we shall have-- n' - a n -s -------- = --- = -1, n' = 0, [therefore] -------- = -1, whence m' - a m m' - a -a = a - m', or m = 2a. The other view of the question is, that F turns once backward in its bearings during each forward revolution of T; whence in Eq. 2 we have-- n' n -------- = --- = -1, n' = -a, m' - a m -a [therefore] -------- -1, which gives -a = a - m', or m' = 2a, m' - a as before. It is next to be remarked, that the errors which arise from applying Eq. I. to incomplete trains may in some cases counterbalance and neutralize each other, so that the final result is correct. [Illustration: PLANETARY WHEEL TRAINS. Fig. 20] For example, take the combination shown in Fig. 20. This consists of a train-arm T revolving about the vertical axis OO of the fixed wheel A, which is equal in diameter to F, which receives its motion by the intervention of one idle wheel carried by a stud S fixed in the arm. The second train-arm T' is fixed to the shaft of F and turns with it; A' is secured to the arm T, and F' is actuated by A' also through a single idler carried by T'. We have here a compound train, consisting of two simple planetary trains, A--F and A'--F'; and its action is to be determined by considering them separately. First suppose T' to be removed and find the motion of F; next suppose F to be removed and T fixed, and find the rotation of F'; and finally combine these results, noting that the motion of T' is the same as that of F, and the motion of A' the same as that of T. Then, according to the analysis of Prof. Willis, we shall have (substituting the symbol _t_ for _a_ in the equation of the second train, in order to avoid confusion): n n' - a 1. Train A--F. --- = 1 = --------; m' = 0, m m' - a n' - a whence -------- = 1, n' = 0, = rot. of F. a n n' - t 2. Train A'--F'. --- = 1 = --------; m' = 0, m m' - t n' - t whence again -------- = 1, t = 0, = rot. of F'. -t