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CHAPTER III.

Theory of the Polar Planimeter.

Instrument Constants.

It will be readily seen from the description of the instrument given in Chapter II, and from the diagram, that certain of the quantities or dimensions thus designated are constant for any given instruments, while others vary and may be given any desired value.

Those dimensions which have a constant value for any given instrument are termed the “Constants” for that instrument or “Instrument Constants” and should be determined with great accuracy.

The “Instrument Constants” for the particular Planimeter we are using have been determined and are as follows:
p = 154.0 L = 208.0
f = 23.0 c = 61.24
b = 7.0

all values being expressed in millimeters.

Of these values p, f, b and L are found by direct measurement of the parts themselves, while the value of c or the circumference of the Wheel W, which is evidently also a Constant for any given instrument, is found by calculation in the following manner:

Having very accurately drawn any plane figure having a known area, such for example as a square or circle having a known area of say 10,000 Sq. Millimeters, we carefully trace that area with any Setting whatever and take the Reading of the Wheel due to tracing that area; this Reading is the value r.

Let us suppose for example that having traced the known area of 10,000 Sq. Millimeters with a setting of say 29.4 the resulting Reading of the Wheel is 1,000.

Since graduation of the Tracer Arm is in one-half millimeters the distance OV represented by the Setting will evidently be 29.4 ÷ 2 = 14.7 millimeters and from Fig. 4, Plate XII.

t = FT = OT – OF = L – (s ÷ 2 + b + f) ... (1)

But by measurement we have for this particular instrument b = 7.0, f = 23.0 and L = 208.0
Hence

t = 208.0 – (14.7 + 7.0 + 23.0) = 208.0 – 44.7 = 163.3 mm.

It will be demonstrated later that

c = A ÷ (r × t)

Hence substituting our known values of A, r and t we have as the desired circumference of wheel

c = 10000 ÷ (1000 × 163.3) = 61.24 millimeters

In the case of the Coradi Compensating Planimeter and one or two other forms of the instrument the Zero of the Tracer Arm graduation is at the Tracer end of the arm and in such a position that the length of Tracer Arm for any operation or value of t is always the same as the Setting divided by 2 which much simplifies this and other calculations.

As a full description of the Polar Planimeter and all its constituent parts has been given in detail in Chapter II no further description of any such part when referred to in the following demonstrations will be necessary; and when any letters or symbols referring to the instrument, its details or factors are used it will be considered that the list of such letters and symbols just given will have been sufficient to admit of the quantity thus referred to being at once recognized.

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