## Theory of the Polar Planimeter.

### 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.