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CHAPTER III.

Theory of the Polar Planimeter.

Revolutions of the Wheel and Their Significance.

The component Rolling and Slipping Motions of Wheel due to movement of the Wheel along any element of its path being understood, the relation of these component motions to the operation of the instrument in tracing any given figure whose area is desired should now be considered.

In Fig. 2 of Plate XII let T'T be the component represented by TS of Fig. 1 of the motion of T while tracing an element of the periphery of a figure whose area is to be measured. Let f be the very small angle at the center subtended by T'T, O'O the arc of the Zero Circle, W'W the path of the Wheel during the movement T'T, and PE a perpendicular let fall on T'F' produced.

The dotted lines PW and PW' drawn from the Pole to the initial and final positions of the wheel for a given movement of T subtend an angle at P equal to the angle f.

The path W'W of the wheel is resolved into the two components W'S and SW; SW representing the distance rolled by the wheel and W'S the distance slipped by the wheel during its movement from W' to W along its path.

Let U be the arc subtending the angle f at a distance of Unity from P.

Since

W'W = Arc × Radius
we have
W'W = U × WP ... (1)

It has already been shown that the distance rolled by the Wheel for any element of its path is equal to the length of that element multiplied by the sine of the angle which that element makes with the axis of the Wheel or from Eq. 1 of Pg. 32:

Distance rolled = l × Sin f ... (2)
Hence
Distance rolled = SW = U × WP × Cos W'WS ... (3)

Since f is very small the angle W'WS can be considered as being equal to the angle PW'E.
But

W'E = W'P × Cos PW'E
= W'P × Cos W'WS ... (4)
Hence substituting in Eq. 3
Distance Rolled = WS = U × W'E ... (5)
But
W'E = p × Cos EF'P' – f = p Cos a – f ... (6)
Hence
Distance Rolled = SW = U × (p Cos a – f) ... (7)
which is the distance rolled by the Wheel, while the Tracer is moved from T' to T along the periphery of Fig. ABT'T in Fig. 1.

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