n 20 n’ 20
  For the wheel F, —–­ = ——­ = ——­, [therefore] n’ = — ——­ a;
                    m 19 -a 19

                    n n’
  For the wheel H, —–­ = 1 = ——­, [therefore] n’ = -a;
                    m -a

                    n 20 n’ 20
  For the wheel K, —–­ = ——­ = ——­, [therefore] n’ = — ——­ a,
                    m 21 -a 21

which corresponds with the actual state of things; all three wheels rotate in the same direction, the central one at the same rate as the train arm, one a little more rapidly and the third a little more slowly.

It is, then, absolutely necessary to make this modification in the general formula, in order to apply it in determining the rotations of any wheel of an epicyclic train whose axis is not parallel to that of the sun-wheels.  And in this modified form it applies equally well to the original arrangement of Ferguson’s paradox, if we abandon the artificial distinction between “absolute” and “relative” rotations of the planet-wheels, and regard a spur-wheel, like any other, as rotating on its axis when it turns in its bearings; the action of the device shown in Fig. 18 being thus explained by saying that the wheel H turns once backward during each forward revolution of the train-arm, while F turns a little more and K a little less than once, in the same direction.  In this way the classification and analysis of these combinations are made more simple and consistent, and the incongruities above pointed out are avoided; since, without regard to the kind of gearing employed or the relative positions of the axes, we have the two equations: 

n’ — a     n
I. -------- = ---, for all complete trains;
m’ — a     m

n’      n
II. -------- = ---, for all incomplete trains.
m’ — a     m

[Illustration:  PLANETARY WHEEL TRAINS.  Fig. 19]

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—­