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CHAPTER III.

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Theory of the Polar Planimeter.

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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.