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A.PPENDIX VIII.

LIMITS OV ERRORS IN SURVEYING. |By W. T. Nicti.n, District Surveyor, Dunedin.] At the present time, when greater accuracy is required in field operations in connection with standard traverses and major triangulation, the writer is of opinion that the limit cf error of the various classes of measurements should be reduced to a uniform standard, based on the mathematical theory of the probability of errors of observation. The following is an endeavour to attain this desirable end, which may be of interest and value to professional surveyors in the Dominion. Pakt 1. Limits oe Eeroes in Subveying. Under the regulations for conducting the survey of land in New Zealand for 1897 the following are the extreme errors allowable : — (1.) Minor triangulation, 2 links per mile. Error in the summation of angles of a triangle, 30". (2.) Closing error of traverses, 4 links per mile. Error of bearing, 3. (3.) Closing error of city traverses, 2 links per mile. These values were revised under the 1908 regulations, and are given as follows : — (1.) Minor triangulation, 6 in. per mile. Error in the summation of angles of a triangle, 20". (2.) Closing error of traverses, 4 links. Error of bearing, 2. (3.) Closing error of city traverses, 1 link per mile. All work having error in excess of those limits requires revising. The degree of accuracy attained in field operations depends on a number of causes, among which arc —weather-conditions ; instability of the ground, as in peat, swamps, and moss growths in forests, and, in town work, the vibration of the traffic, &c. The carefulness and accuracy of the surveyor and the chainmen are largo factors in the accuracy of any survey. The principal factor, however, affecting the accuracy of a survey is dependent on the instruments and apparatus used in the performance of the work, and this factor alone will be the subject of the following theoretical investigations : — A Determination of the Closing Error in Traverses made nil It a Sin. Transit Theodolite, and a long Steel Band. Investigations of the effects of errors in surveying require the application of the results derived from the theory of errors. One of the most important results is the probability curve, or curve of errors, the equation of which is— , y= k c "x ' (O This is termed the exponential law of errors, k and h being constants, and c the base of the Napierian system of logarithms. Prom equation (1) a criterion of the degree of uncertainty of the result of a number of measures is deduced. The criterion of the degree of uncertainty of the result of a series of equally good measures or observations has three distinct definitions, — (1.) The mean error, or the average error, is defined as the arithmetic mean of the separate errors taken all with the same sign. (2.) The error of mean square is defined as the square root of the arithmetic mean of the squares of the individual errors. (3.) The probable error- is such that there are as many errors of greater magnitude as there are of smaller magnitude. The following table, from Airy's " Theory of Errors," shows the connection between the mean error, the error of mean square, and the probable error, and, when one is known, by use of the table it can be converted to either of the other two. Proportions of the different constants, —

These three criteria—namely, the mean error (E m ), the error of mean square (7o? s ), and the probable error (E v ) —are equally good from a theoretical standpoint, and in selecting one of them for the purpose of testing the accuracy of a field traverse preference is given to that which is easiest to compute —viz., the mean error.

n terms of modulus "n terms of mean error "n terms of mean error of square n terms of probable error ... Modulus. 1-000000 1-772454 ... I 1-414214 ... i 2-096665 Mean Error, 0.564189 1000000 0-797885 1-182910 Error of Mean Probable Square. Error. 0-707107 0-476949 1-253314 0-845369 I 000000 0-674506 1-482567 1-000000