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CHAPTER III.

Theory of the Polar Planimeter.

Constants— Their Derivation and Use.

While the demonstrations and descriptions thus far given cover all the factors involved in the ordinary methods of Planimeter measurements, in a succeeding chapter is given a method by which the areas of figures too large to be measured by the ordinary methods can be accurately determined.

It will be seen from the description of the method referred to, that having placed the Pole of the instrument on the inside of the large figure to be measured and traced the given figure, the resulting Reading must be added algebraically to a certain “Constant” before the desired area can be obtained.

The derivation and significance of these Constants are very easily understood in the light of the previous discussion of the Zero Circle and its relation to the theory of the Planimeter.

In the discussion referred to it was shown that the distance rolled by the wheel or the “roll” of the wheel when multiplied by the length of the Tracer Arm gave the area of that portion of the figure included between the path of the Tracer, the arc of the Zero Circle, and the radii drawn to the initial and final positions of the Tracer.

It is evident from this that if the area of the figure to be measured is a large one, and we place the Pole near the center of the given area and trace its periphery, the resulting Reading of the Planimeter, when multiplied by the length of the Tracer Arm, will give the total area of the figure traced, minus the area of the Zero Circle. In other words, the instrument will evidently measure the difference between the area of the figure thus traced and the area of the Zero Circle, and hence the area of the given figure will be the algebraic sum of the area recorded by the Wheel and the area of the Zero Circle.

The Radius of the Zero Circle has already been shown to be

R = sqrt(p² + 2ft + t²) ... (1)
and its Area is
pi R² = pi (p² + 2ft + t²) ... (2)

In finding the actual area of any given figure by this method the constant used would evidently have to be the area of the Zero Circle, expressed as the number of Vernier Units which would be recorded by the Planimeter if the Constant Circle were traced with the given Setting for the operation in which it occurs.

In other words, the Constant for the measurement of any area drawn to a given scale is evidently the difference between the number of Vernier Units which would be recorded could the entire area have been measured by the Planimeter with the Pole on the outside of the given figure.

It is readily seen that the actual area of the Zero Circle must change for any change in t, since p and f in the expression p² + 2ft + t² are constant for any given instrument and t is variable, and hence there must be a corresponding value of the “Constant” for every given “Setting.”

In the description given in Chap. IV of the method of measurement of large areas with the Pole on the interior of the given area, the significance, derivation and use of the “Constant” is so fully described that further discussion of the subject at this point is unnecessary, although the theory and application of the Constant should be easily understood from what has already been said here on the subject.

In fact, in all the following descriptions of the use of the Planimeter in all its many practical applications, the relation of the theory of the particular operations described is made clear, thus supplementing the discussion given here of the theoretical considerations involved.

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