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Theory of the Polar Planimeter.

Range of the Instrument.

In Eq. 5 it is evident that c is a Constant for any given Instrument, and A is a Constant for any given operation.

For the particular instrument we are using we have already found— (Pg. 31)— c = 61.24 mm. Assuming A = 10000 Square Millimeters, and substituting in Eq. 5, we have

r × t = 10000 ÷ 61.24 = 163.292 ... (6)

This value of r × t is evidently a Constant for this particular instrument for the given value of A, which we have taken as 10000 Sq. mm.

Eq. 6 shows also that since r × t is a Constant for any given instrument and for any given area, r is dependent on t for its value, and any change in t produces a corresponding change in the value of r.

On examining the graduation of the Tracer Arm of our Planimeter, it will be seen that the minimum value which the construction of the instrument allows being given to t for the instrument is about 60 mm, while the maximum value is about 180 mm.

Substituting these values respectively in Eq. 6 and reducing, we have as the resulting No. of Vernier Units

1000 r  = 163292 ÷ t = 163292 ÷ 60 = 2721.5 ... (7)
1000 r = 163292 ÷ t = 163292 ÷ 180 = 907.1 ... (8)

These Equations show that by changing the length of the Tracer Arm we can make the given Planimeter record any desired number of Vernier Units between 2721.5 and 907.1 Vernier Units while tracing an actual area of 10000 Sq. Millimeters; or in other words, that the range of the given Planimeter for an area of 10000 Sq. Millimeters is from 907.1 to 2721.5 Vernier Units. Since 10000 Sq. Millimeters equals 15.5 Sq. Inches the range of the instrument for 1 Sq. inch would evidently be from 2721.5 ÷ 15.5 = 175.55 to 907.1 ÷ 15.5 = 58.52 Vernier Units.

The reason why, both in calculations of the Tables and in all or most of the mathematical demonstrations given throughout, the value usually assigned to A in any given operation is 10000 Sq. Millimeters, or its equal 15.5 Sq. inches instead of 1 Sq. inch is already stated in the “Explanation of Tables,” but will be explained in fuller detail later in the discussion.

It should be remembered that r is the distance rolled and recorded by the Wheel for any given tracing or operation, and can be expressed either as the number of complete revolutions and fractions of a revolution, or as the number of Vernier Units recorded, since the circumference of the Wheel is divided into 1000 equal parts or Vernier Units. In any event, it is always the “Reading” of the Planimeter for the given tracing or operation, and the change from Revolutions and fractions to Vernier Units is made by simply moving the position of the decimal point.

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