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XXVIII. — An Investigation into the Effects of Errors in Surveying. By Henry 

 Briggs, B.Sc, A.R.S.M. Communicated by Principal A. P. Laurie. 



(MS. received January 7, 1911. Read February 6, 1911. Issued separately April 18, 1911.) 



This paper discusses the effects of errors in linear and angular measurements on the 

 accuracy of surveys. 



It is necessary to state at the outset that, as the investigation is based on the theory 

 of probability, the conclusions drawn are only advanced in the hope that they will serve 

 as guides in practical surveying, and must not be taken as of rigid precision. Perhaps it 

 is superfluous to state that there is no method of determining beforehand the error which 

 will accumulate in any survey ; the utmost that can be done is to assign some mean value 

 to the error which will serve as a useful criterion to which actual errors can be referred. 



In dealing with probability far more difficulty is encountered in laying down trust- 

 worthy premises than in building mathematical structures upon them, and it would 

 seem that the most satisfactory method of testing whether the premises are sound is to 

 compare one's results as often as possible with those derived from experience. At 

 several points the results in this paper have been so tested, and indeed some of them 

 have turned out to be nothing else than algebraic representations of facts which have 

 long been known in a general and more or less shadowy way to surveyors. In the last 

 section of the paper the practical bearing of the results is emphasised, partly to show to 

 what extent they conform with empirical conclusions. 



The Average Error. 



As it is impossible to deal with actual errors in a general investigation, choice has to 

 be made of some form of mean error as a representative value. The average error, or 

 average deviation as it would better be called, is selected for the purpose, firstly, because 

 it is the standard of comparison most serviceable in a discussion of the relative accuracy 

 of different results and processes ; secondly, because it is more readily determined in 

 practice than either the probable error or the mean-square error, and also because it is 

 more easily understood by those unfamiliar with the theory of errors. 



The average error in a series of equally trustworthy measurements of the same thing, 

 whether a length or an angle, is defined as the arithmetical mean of the separate errors, 

 taken either all with positive or all with negative signs.* 



* To write the plus-or-minus sign before average errors, as is done throughout the paper, is unorthodox. An 

 average error is usually denned as the arithmetical mean of the separate errors taken regardless of their signs, and is 



expressed in books as \ +v ' It is, however, equally logical to express it as ^ — ^, and hence it appears to have the 



same right to the double sign as, say, the probable error. Moreover, to give an average error the positive sign only, 

 and then to apply the usual theory of errors to it (as is generally done) seems to the writer inconsistent, since there is, 

 in such an application, a tacit admission that the error is minus as often as plus 



TRANS. ROY. SOC. EDIN., VOL. XLVII. PART IV. (NO. 28). 125 



