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.
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