REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 93 



that formula assumes it to be + -L - (21), or rather it 



n 



is on the assumption that the probable error of a residual is deter- 

 mined by the probable value of (21), that n — 1 appears as a 

 denominator (instead of n) in the formulae. Hence it is evident 

 that if these denominators were determinable in strict accordance 

 with the probability of the cases to which the formulae are applied 

 they would appear as some quantity greater than n but less than 

 n — 1, and in some instances it will be desirable, as contributing 



more approximate results, to employ '675 v ...(22), or *845 — ■ 



(23), rather than (18) or (19). This desirability may often 



be decided a priori. 



As indicating the illegitimacy of finding the probable error of 

 the final result by dividing that of a single measure by the square 

 root of the number, and exhibiting at the same time the algorithm 

 of a truer process, an example of nineteen repetition measures of 

 an angle is given, taken advisedly from the records of observations 

 with small theodolites, in order to display the extraordinary pre- 

 cision of results obtained by the " repeating " method. 



In I., Column 1 shews the number of " repeats "; 2 the suc- 

 cessive values of the mean of the readings of verniers A and B ;. 

 3 the separate measures of the angle as obtained by subtracting 

 each reading from that next following ; 4 the residuals v obtained 

 by subtracting the arithmetic mean (or what is the same thing 

 the final reading divided by 19, the number of repeats) from each 

 measure of the angle ; 5 the squares of these residuals v 2 ; 6 the 

 several readings, in column 2, divided by the number of repeats- 

 Column 7 after the double line shews the residuals v formed by 

 subtracting the more probable value of the angle as found by the 

 method exhibited in II.; and 8 their squares v' 2 . 



In II., Column 1 shews the readings taken to form the series 

 of ten repetition measures ; 2 the total angle found by subtracting 

 from 10, 1 from 11, etc.; 3 the value of the angle obtained by 

 dividing the total angle by the number of repetitions, 10; 4 the 

 residuals formed by subtracting the arithmetic mean of these 

 different values from them individually ; and 5 the squares of the 

 residuals. 



By (22) -675 v( 16 ^ -) = + 6-3". By (23) -845 13 ^ 7 = + 6-0". 



= 4- 1 -4.0" * 

 — V19 



+ 140".* (See I. on following page.) 



* This would be the probable error of the mean of nineteen such 

 measures if they were independent and collateral, which however they 

 are not. In regard to the expression of results to so (in this instance) 

 excessive a degree of precision, see a note later on concerning this in 

 particular. (See I. on following page.) 



