m -a
n 20 n' - a 1
" " " K, --- = ---- = ---------, [therefore] n' = + ---- a.
m 21 -a 21
The paradoxical appearance, then, consists in this, that although the
drivers of the three last wheels each have the same number of teeth,
yet the central one, H, having a motion of circular translation,
remains always parallel to itself, and relatively to it the upper one
seems to turn in the same direction as the train-arm, and the lower in
the contrary direction. And the appearance is accepted, too, as a
reality; being explained, agreeably to the analysis just given, by
saying that H has no absolute rotation about its axis, while the other
wheels have; that of F being positive and that of K negative.
[Illustration: PLANETARY WHEEL TRAINS. Fig. 18]
The Mechanical Paradox, it is clear, may be regarded as composed of
three separate trains, each of which is precisely like that of Fig.
16: and that, again, differs from the one of Fig. 15 only in the
addition of a third wheel. Now, we submit that the train shown in Fig.
17 is mechanically equivalent to that of Fig. 15; the velocity ratio
and the directional relation being the same in both. And if in Fig. 17
we remove the index P, and fix upon its shaft three wheels like E, G,
and I of Fig.
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