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CHAPTER III.

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Theory of the Polar Planimeter.

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Symbols and Notation.

In order to make all demonstrations and discussions clearer and the mathematical
treatment of the theory and applications of the Instrument connected and
uniform throughout, certain letters and symbols will be assigned to represent
the different parts or dimensions of the Planimeter, as well as other factors,
constants and other values entering into either the descriptions
of the instrument and its operations and applications or the mathematical
discussion connected with such descriptions, so that when any such
letter or symbol is employed or occurs in any portion of the subsequent
treatise it will be at once understood as representing the quantity or
value thus assigned it.
In the following demonstration of the general theory of the Polar Planimeter
and the relation existing between its various parts, and in fact throughout
all the subsequent discussion of the instrument and its practical applications,
the different parts and dimensions of the Planimeter will then be denoted
by the following letters, which are also used in all
the diagrams and drawings used to illustrate the descriptions— it being remembered
that all dimensions are given in millimeters and that the unit of graduation
of the tracer arm is one-half millimeter.

Referring then to Fig. 4, Plate XII, which
represents the principal parts of the Polar Planimeter, these parts and
their dimensions will be designated as follows:

p = PF = Length of Polar Arm.

f = FW = Distance between Wheel and Carriage Pivot.

L = OT = Total length of Tracer Arm.

b = VW = Distance between Carriage Vernier Zero and Wheel.

s = OV = Setting.

t = FT = OT – (OV + VF) = Length of Tracer Arm.

r = Reading of Wheel = No. of Vernier Units recorded for tracing any
area A with any given value of s or t.

c = Circumference of Wheel W.

x = Value of Relative Vernier Unit for any given operation.

y = Value of Corresponding Actual Vernier Unit.

R = Radius of Constant Circle.

A = Area traced with length of Tracer Arm t and which gives reading
r for that tracing.

C = Constant for inside position of Pole.