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CHAPTER III.

Theory of the Polar Planimeter.

Settings— Their Determination and Use.

Having just shown the relation existing between the values of r and t for our particular instrument and the maximum and minimum values attainable for r for tracing an actual area of 10,000 Sq. Millimeters, we are able to explain the method by which we determine the value that must be given to t in order to obtain any desired value of “r” for the same area; in other words to find the length of Tracer Arm necessary to cause the Wheel to record any desired number of revolutions while tracing an area of 10,000 Sq. Millimeters.

Eq. 4 of our Fundamental Equations is as already given

t = A ÷ (c × r) ... (4)

Substituting the value of c already formed for our particular instrument and making A = 10,000 Sq. mm we have

t = A ÷ (c × r) = 10,000 ÷ (61.24 × r) = 163.292 ÷ r ... (5)

Since r is the number of revolutions which we desire the Wheel to record while tracing the given area of 10,000 Sq. Millimeters, it is evident form Eq. 5 that the required length of Tracer Arm is obtained by dividing the Constant 163.292 by the desired reading or value of “r.”

The meaning of the term Setting together with the method of adjusting the Planimeter to any given Setting has been fully described in the preceding chapter and a glance at Fig. 1 of Plate III will at once show the relation existing between any given Setting and the corresponding Length of Tracer Arm for any given instrument.

To determine this relation for the particular Planimeter we are using we make use of the skeleton diagram of the instrument shown by Fig. 4 of Plate XII, which shows the essential parts properly lettered, together with the instrument Constants and other dimensions determined either in the manner already explained for finding those Constants, or by direct measurement of the instrument in question.

It is to be remembered that the unit of graduation of the Tracer Arm, being a half Millimeter the value of any Setting is expressed in that unit and hence the actual distance is always one-half of the distance as expressed by the value given for the Setting, or in other words the actual distance S is Setting ÷ 2.

Referring to the diagram it is evident that

t = FT = OT – (OV + VF)
But for this instrument
OT = 208 Millimeters
OV = S
VF = b + f = 7 + 23 = 30 Millimeters
Hence
t = 208 – (S + 30) = 178 – S
from which
S = Setting ÷ 2 = 178 – t ... (6)

Eq. 6 shows that after having found by Eq. 5 what value of t is required in order that the Planimeter shall record the desired number of Revolutions while tracing an area of 10,000 Sq. Millimeters, by substituting this value of t expressed in Millimeters in Eq. 6 and solving we at once obtain the Setting to which the Planimeter must be adjusted in order that the tracer arm shall have the length required.

By substituting for t in Eq. 6 its value as expressed in Eq. 5 we have as the General Equation for finding the Setting required to give any desired value of r for the area of 10,000 Sq. Millimeters,

Setting = (178 – 163.292 ÷ r) × 2
or
Setting = 356.6 – 326.584 ÷ r ... (7)

It is evident that the Setting necessary for any instrument to cause it to record any desired number of Vernier Units for any given area can be found by trial by repeated tracings of the given area with different lengths of tracer arm, but except as a check in any given case the method by calculation as just described is much quicker and easier of application.

To illustrate the application of formulae just deduced let us suppose that we desire to find what Setting our given Planimeter must be adjusted to in order that the instrument should record say 1,550 Vernier Units when tracing an actual area of 10,000 Sq. Millimeters.

Eq. 7 for this particular Planimeter is as just shown

Setting = 356.6 – 326.584 ÷ r ... (7)

Since the desired value of r is 1550 Vernier Units or, when expressed in Revolutions of Integrating Wheel, 1.55 Revolutions, we substitute this value of r in Eq. 7 which gives

Setting = 356.6 – 326.584 ÷ 1.55 – 356.6 – 210.61 = 145.9
which is the Setting desired.

The reason why the actual area of 10,000 Sq. Millimeters or its equal 15.5 Sq. Ins., is always taken in this calculation and in fact in the calculation of all factors in the Tables is that not only does it simplify the calculation, but it is usually one of the constant areas found on the Test Plate accompanying most Planimeters and hence makes the checking of the calculations a very simple matter.

Thus, in the example just given, having obtained the value of the desired Setting in the manner shown we at once adjust the Planimeter to 145.9, the Setting in question, and trace the given area (10,000 Sq. Millimeters) with the aid of the test plate. In this way the accuracy of the calculation is at once tested.

The Cols. marked “Area” and “Reading” under the heading “Test Plate” in all the Tables in connection with the values given in the “Relative” and “Actual Vernier Units” columns evidently give at once a very accurate test, both of the accuracy of the Setting for any given scale and of the adjustment and condition of the Planimeter and these should be used in connection with the Test Plate area of 10,000 Sq. Millimeters to check the accuracy of the Planimeter before each use of the Planimeter.

On examining the diagram given on Plate II, it is readily seen that the curve BB is simply the graphical representation of Eq. 7 for the various values of r for an area of 10,000 Sq. Millimeters between the limits of the given instrument; the curve being the various values of the Settings and having for its ordinates the values of r while the abscissa are corresponding values of S.

The determination for any given Planimeter of the Setting necessary to cause the instrument to record a certain number of desired Vernier Units, when tracing a given area is greatly facilitated by the use of a diagram similar to that given in Plate II.

As this diagram can by applying a proper connection be used for any Planimeter similar to the one we are discussing the method of preparing the diagram will be described at this point, and its manner of use described. The size of the diagram having been decided upon, it is ruled in squares in the manner shown and the range of the instrument having been determined, the values of r due to tracing a constant area of 10,000 Sq. Millimeters between the maximum and minimum values of the length tracer arm are lettered in a vertical column at the end as shown.

The upper base of the diagram is taken as representing to some scale of equal parts, the various divisions and subdivisions of the Tracer Arm graduation and lettered or numbered accordingly.

We have seen that the Setting necessary to give any desired Reading or value of r for the particular Planimeter we are using is given by Eq. 7 just demonstrated and is

Setting = 356.6 – 326.584 ÷ r ... (7)

If now we take the various values of r between the limits of the range of our instrument and substitute them successively in Eq. 7— solving each such resulting Equation— we can readily plot the resulting values of the respective Settings on the diagram— marking each such location by a dot— and by connecting these located dots by a curved line we shall have the curve BB.

It is evident that this curve is the curve of Settings for the given Planimeter plotted to co-ordinates in which any ordinate to the curve is a number of revolutions and its corresponding abscissa is the distance represented by the Setting (or the distance S) necessary in order that the wheel shall record that number of revolutions when tracing an area of 10,000 Sq. mm.

In the diagram the curve BB is for use with the Planimeter we are using while the curve AA is the curve of Settings for those Planimeters like the Compensating Planimeter which has the Zero of its Tracer Arm graduation at or near the Tracer. The upper base is numbered for use with the curve BB while the lower base is for use with the curve AA.

In plotting if the curve of the even numbered revolutions— 1,000, 1,100, 1,200, &c. are calculated and plotted and the curve constructed from their values the intermediate values will be found quite accurate enough for any but the most accurate operations.

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