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CHAPTER III.

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Theory of the Polar Planimeter.

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Relation of Roll of Wheel to Area Traced.

In Fig. 3 of Plate XII, let ABC be any given
figure of whose area pq is an element. Let P, W and T be the Pole, Wheel
and Tracer respectively of the Planimeter, and let the Tracer move across
the width of the element of p, the width of the element being very small.
Let OO be an arc of the Zero Circle, the upper
width of the element being coincident with it at the point q. Let U be
the small angle at the Pole subtended by the width of the Element pq, while
the parts of the Planimeter are designated by the letter assigned them.

Then

Area pq = ½ U (p² + t² + 2 pt Cos *a*)
½ U (p² + t² + 2 ft)

= Ut (p Cos *a* f) ... (1)
But the area pq is the area included between the portion of the periphery
of the figure traced, the arc of the Zero Circle, and the radii drawn from
the Pole to the beginning and end of the line traced by T. We have already
found that the distance which the Wheel would roll and record after tracing
the width of the element at b is:
Distance rolled = U (p Cos *a* f) ... (2)
Comparing Eqs. 1 and 2, it is seen that *the area of the element is
equal to the distance rolled by the wheel for the given tracing multiplied
by the length of the tracer arm t.*

As this is true for *any* element of area of the figure traced,
it must be true of *every* element, and hence the total area of the
figure outside the arc of the zero circle.

It will require but little consideration to show that the instrument
deals in like manner with that portion of the traced figure included *within*
the Zero Circle, and that the rolling of the Wheel resulting from the tracing
of the periphery of any given closed figure when multiplied by the length
of the Tracer Arm gives at once the area of that figure.

If c is the circumference of the Wheel and r the number of revolutions
made by the Wheel during any given tracing, it is evident that the distance
rolled by the Wheel, which we have also referred to as the roll of the
wheel, is c × r. Hence, denoting the area of any figure by A, and
the length of Tracer Arm by t, we have

A = c × r × t
which is what we have already termed the General Equation
of the Polar Planimeter, and from which is at once obtained the General
Principle of the instrument.