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CHAPTER III.

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Theory of the Polar Planimeter.

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The Zero Circle.

From the discussion just given of the Rolling and
Slipping components of the travel of the wheel W,
it is seen that there will evidently be some path or line along which the
Wheel can be moved without producing any resulting rolling of the Wheel—
the movement of the Wheel along this path being one of slipping only.
If we so adjust the positions of the two arms of the Planimeter that
a line drawn from the Pole P to the Wheel W shall be at right angles to
the Tracer Arm, and with the arms fixed in this position we cause the entire
instrument to revolve about P as a center, it is seen that the Wheel W,
the Joint F, and the Tracer T, will each describe arcs of concentric circles having
P as a common center. It is also readily seen that since the axis of the
Wheel is at every point of the circle thus traced by the point of tangency
of the Wheel at right angles to the element of the circle at that point,
there cannot be any rolling of the Wheel whatever so long as the arms retain
this position with respect to each other. Of the three circles thus formed
the circle described by the point of tangency of the Wheel is called the
“Line of Slipping,” that described by the Pivot Joint F is called the “Directrix,”
while the circle traced by the Tracer during this revolution is called
the *“Zero Circle,”* as shown in Fig. 2 of Plate
III.

The “Zero Circle” can then be defined as being a line along which the
Tracer can be moved without producing any rolling of the Wheel whatever.

Fig. 1 of Plate III shows a plan of the Planimeter
when the Arms are in position to cause the Tracer to describe the Zero
Circle, while Fig. 2 of the same Plate shows a lettered skeleton diagram
of the instrument when in this position.

Referring to Fig. 2, just mentioned, it is readily shown that the Radius
of the Zero Circle is

R = sqrt(p² + 2ft + t²)
while
*a* = Cos^{–1} (f ÷ p)
On examining the graduation of the wheel it will be seen that when the
wheel revolves in the direction of a right-handed screw the revolution
will be positive, while revolution in an opposite direction would be backward
or negative.

When the Tracer moves about P as a center in the direction of the hands
of a watch the *direction* of the tracing is *positive*, while
motion an opposite direction *negative*.

*Outside* the Zero Circle, tracing in a *positive direction*
gives a *positive rolling* of the Wheel while tracing in a *negative
direction* gives a *negative rolling*.

*Inside* the Zero Circle, tracing in a *positive direction*
gives a *negative rolling* of the Wheel, while tracing in a *negative
direction* gives a *positive rolling.*