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Not only is a knowledge of the degree of accuracy attainable in these operations of the greatest possible value as furnished proof of the reliance which may be placed in all results of measurements thus obtained, but they serve the added purpose of showing the almost incalculable value of the aid these instruments are capable of rendering in every form of Engineering work and giving them the high place in the list of the Engineer mechanical assistants for which that accuracy and adaptability so eminently fits them.

While this subject has engaged the attention of a number of investigators, and many experiments have been made at times with more or less valuable results, perhaps the most valuable conclusions reached have been those due to a series of experiments conducted by Prof. Lorber while investigating the subject and reported by Prof. Shaw in his admirable paper read before the Institution of Civil Engineers of England and reprinted from their Proceedings.

In conducting these tests no attempt whatever was made to have the conditions during the experiments any more favorable than would be the case in the ordinary use of the instrument, so that the results obtained are those which may confidently be expected in every case in which the Planimeter is understandingly used and in compliance with the conditions for accuracy already given in a previous chapter.

It should be stated that the results obtained and reported were from a single tracing of the figure measured and not the average of a number of repetitions of the same measurement, and that the results are the average of nine different Planimeters, thus making them equivalent to the results of an average instrument.

From the series of experiments referred to Prof. Lorber concludes that “the different angles at which the measuring roller of the Polar Planimeter acts has little effect upon the results.” He also found that taking one turn of the measuring roller, as + 100 Sq. Centimeters, the average error in the reading was only from 0.00075 to 0.0013, according as the center of rotation or the pole was without or within the area to be measured.

As the results of all of his experiments, Prof. Lorber offers the following empirical formula as giving the average error for the different instruments for average conditions:

n = Reading of Integrating WheelApplying to the various forms of Planimeters, of which we have given descriptions in the previous chapters, and introducing the values furnished by experiment for the factors k and n, we have as the error of the different classes of Planimeter

F = Actual Area to be measured

d F_{n}= Error in result expressed in terms of the Area

dF = 0.00126 f + 0.00022 sqrt(F f) for the Polar Planimeter

dF = 0.00069 f + 0.00018 sqrt(F f) for the Precision Polar Planimeter

dF = 0.0009 f + 0.0006 sqrt(F f) for the Rolling (Coradi) Planimeter

The degree of accuracy as determined by the application of these
formulas to a given operation may be inferred from a trial measurement
taken with the Coradi Rolling Planimeter where
f = 100, which gave the relative error of that instrument for the given
instrument as

dF ÷ F = 1 / 13330

a degree of accuracy unattainable by any other method of measurement.

The actual results attained by the application of Prof. Lorber's formula
is best shown in the following table, prepared by him as showing the results
of his investigations:

Actual Area
Measured |
Absolute Error
in Result |
Relative Error
in Result |
|||||

F cm² | d F_{n} cm² |
f = 100 | cm² | Per cent | |||

200 | 0.157 | 1 in. | 1274 | 0.0785 | |||

100 | 0.148 | 1 in. | 682 | 0.148 | |||

50 | 0.141 | 1 in. | 355 | 0.282 | |||

20 | 0.135 | 1 in. | 148 | 0.675 | |||

10 | 0.133 | 1 in. | 75 | 1.330 | |||

5 | 0.131 | 1 in. | 39 | 2.620 |

The above results are from single, not repeated, tracings of the given area, and from the table it is seen that

1. Absolute Errors differ but little and are not proportional to the Actual Area traced.

2. The Relative Error diminishes rapidly as the Area increases and are almost inversely proportional to the increase in Area.

The results of similar experiments made by other investigators have been stated in many different ways, most of them however being the results of experiments involving some one particular application, so that the empirical formulas as given by Prof. Lorber probably is more general and covers the subject more completely than do any or all of the others.

As the actual and relative degrees of accuracy which may be expected in any particular practical application of the Polar Planimeter has been stated in connection with the description of that particular application in the preceding chapters, it need not be repeated here, but enough has been said to prove the truth of the assertion made of the Planimeter at the beginning of our discussion that “a knowledge of its invaluable capabilities and of the enormous saving in time and labor effected by its use is the only requisite to make it the invaluable and inseparable co-laborer of the Engineer in almost every detail of his professional life.

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