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