﻿IN THE METHOD OF LEAST SQUARES. 211 



bility of the three systems, we obtain first the residual errors of the original equa- 

 tions as follows : — 



Residual Errors of Original Equations. 



I. (Revised solution.) II. III. 



// // II 



1 — 125.45 



2 — 9.14 



3 + 86.62 



4 — 53.78 



5 + 32.12 



6 4- 22.92 



7 + 21.00 



8 + 35.51 



9 + 149.37 



10 — 169.77 



11 + 66.95 



The sums of the squares of these errors are 



I. 



Q = 85091 



The probable value of a residual error for one of the original equations — since there 

 are eleven equations * and six indeterminates — will be obtained from the expression 



fi = 0.67459 \ — - — , 

 Nil— 6 



giving the values 



I. II. III. 



(47.) A« = ± 88".00 ± 89".77 ± 88".06. 



The agreement of fi in I. and III. is a conclusive proof of the equivalency of the two 

 solutions, notwithstanding the freedom which has been exercised in the choice of fac- 

 tors for the latter. In the case of II., the discrepancy amounts to - of /x ; but it will 

 be noticed that the factors actually employed Avere based upon a system of representa- 

 tion considerably less exact than the series (2). It might easily be shown that this 

 difference would have been reduced to less than one fifth of its present amount if the 

 numbers for the factors had been selected from (2), agreeably to the conditions by 

 which II. has been defined. 



* Equation 9„ having been excluded from each of the solutions. 



