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

Of these results, the first is explicable as being the absolute rotation of F, but the second is not; and it will be readily seen that the former would have been equally absurd, had the axis LL been inclined instead of vertical.  But in either case we should find the errors neutralized upon combining the two, for according to the theory now under consideration, the wheel A’, being fixed to T, turns once upon its axis each time that train arm revolves, and in the same direction; and the revolutions of T’ equal the rotations of F, whence finally in train A’—­F’ we have: 

n         n’ — t
3. --- = 1 = --------; in which t = 0, m’ = a,
m         m’ — t

n’ — 0
which gives    --------- = 1, or n’ = a.
a — 0

This is, unquestionably, correct; and indeed it is quite obvious that the effect upon F’ is the same, whether we say that during a revolution of T the wheel A’ turns once forward and T’ not at all, or adopt the other view and assert that T’ turns once backward and A’ not at all.  But the latter view has the advantage of giving concordant results when the trains are considered separately, and that without regard to the relative positions of the axes or the kind of gearing employed.  Analyzing the action upon this hypothesis, we have: