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
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:
In train A--F:
n n' n'
--- = 1 = --------; m' = 0, [therefore] ---- = 1, or n' = -a;
m m' - a -a
In train A'--F':
n' n' n'
--- = 1 = --------; m' = 0, [therefore] ---- = 1, or n' = -t;
m m' - t -t