## Theory of the Polar Planimeter.

### Analysis of Movements of Integrating Wheel.

As the first step in the analysis of the theory governing the construction and operation of the Polar Planimeter it will be necessary to determine the effect on the wheel W of any motion of the Tracer T.

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 Rolled = l × Sin a ... (1)
Distance Slipped = l × Cos a ... (2)
When a = 0°,
Distance Rolled = l × 0 = 0.
and
Distance Slipped = l × 1 = l.
When a = 90°,
Distance Rolled = l × 1 = l.
and
Distance Slipped = l × 0 = 0.