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It is readily seen from the construction of the instrument that any
motion of T must be either a motion of revolution of T about F as a center
with a corresponding variable value of *a*, a motion of T about P
as a center with a fixed value of *a*, or a combination of the two
motions.

Thus, in Fig. 1 of Plate XII, in tracing the
area of the figure shown, the motion of the Tracer while tracing the element
of its path T'T may be considered as being composed of two component motions,
T'S and ST. Of these components T'S is described by a motion of T about
P as a center with a constant value of *a*, while ST is formed by
a motion of T directly toward the Pole P with a corresponding variable
value of *a*.

It is evident that each of these component motions will have an effect
on the Wheel W, the kind and extent of the motion being dependent on the
direction and length of the element T'T and hence on the value of the angle
*a*.

In tracing any figure such as ABT'T it is seen that when the periphery
of the entire figure has been passed over by the Tracer, the Tracer having
returned to the point of beginning, that during the tracing the Tracer
has moved just as much towards P as it has moved away from it and hence
any revolutions of the Wheel due to movement *toward* P are neutralized
by the same number of revolutions in an opposite direction due to movement
*away*
from P for that tracing:

This shows that the number of revolutions recorded by the Wheel during any given tracing of a closed figure are due entirely to the motion of T about P as a center.

In considering the path of the Wheel W due to motion imparted to the
Wheel by motion of the Tracer T, it is seen that *when the direction
of the path in any given case is at right angles to the axis of the Wheel,*
the Wheel will move along that path by *rolling: When the direction of
the path of the Wheel is in the direction of the axis of the Wheel,*
the Wheel moves along the path by *slipping* and *when the direction
of the Wheel's path is between these two directions* the Wheel moves
along the path *partly rolling, party slipping*, the amount of roll
or slip in any given case being dependent entirely on the angle which the
given path makes with the axis of the Wheel.

Denoting the angle made by any element of the Wheel's path with its
axis by *a* and the length of the element by *l* we have for
that element,

Distance Slipped =

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