*Project Gutenberg*. Public domain.

By Prof. C.W. MACCORD, Sc.D.

[Illustration: PLANETARY WHEEL TRAINS. Fig. 14]

It has already been shown that the rotations of all
the wheels of a planetary train, relatively to the
train-arm, are the same when the arm is in motion
as they would be if it were fixed. Now, in Fig.
14, let A be the first and F the last wheel of an
*incomplete* train, that is, one having but one
sun-wheel. As before, let these be so connected
by intermediate gearing that, when T is stationary,
a rotation of A through *m* degrees shall drive
F through *n* degrees: and also as before,
let T in the same time move through *a* degrees.
Then, if *m’* represent the total motion
of A, we have again,

m’ = m + a, or m = m’ — a.

This is, clearly, the motion of A relatively to the fixed frame of the machine; and is measured from a fixed vertical line through the center of A. Now, if we wish to express the total motion of F relatively to the same fixed frame, we must measure it from a vertical line through the center of F, wherever that maybe; which gives in this case:

n’ = n + a, or n = n’ — a.

but with respect to the train-arm when at rest, we have:

ang. vel. A n ------------ = ---, whence again ang. vel. F m

n’ — a n ------ = --- . m’ — a m

This is the manner in which the equation is deduced
by Prof. Willis, who expressly states that it
applies whether the last wheel F is or is not concentric
with the first wheel A, and also that the train may
be composed of any combinations which transmit rotation
with both a constant velocity ratio and a constant
directional relation. He designates the quantities
*m’*, *n’*, *absolute revolutions*,
as distinguished from the *relative revolutions*
(that is, revolutions relatively to the train-arm),
indicated by the quantities *m*, *n*:
adding, “Hence it appears that the absolute revolutions
of the wheels of epicyclic trains are equal to the
sum of their relative revolutions to the arm, and
of the arm itself, when they take place in the same
direction, and equal to the difference of these revolutions
when in the opposite direction.”

In this deduction of the formula, as in that of Prof. Rankine, all the motions are supposed to have the same direction, corresponding to that of the hands of the clock; and in its application to any given train, the signs of the terms must be changed in case of any contrary motion, as explained in the preceding article.

And both the deduction and the application, in reference to these incomplete trains in which the last wheel is carried by the train-arm, clearly involve and depend upon the resolving of a motion of revolution into the components of a circular translation and a rotation, in the manner previously discussed.