﻿IN THE METHOD OF LEAST SQUARES. 197 



(28) may easily be extended to the analogous cases of the second members of the 

 equations 



P' (x- — X,) + Q (2/ — i/i) + = e. 8^ + e'. 8/3' + 



P" (x — X,) -\- Q^ {y — !/,) + = fi8y + e', 8/ + 



so that 



(?) (r) 



(29.) g*— iP>-^, g^ — iry^-L, 



^ ' (Q) (R) 



give the limits within which the proposed changes of the factors 13, ^' y, -y' 



■will not increase the probable sums e, 8 /3 -(- e'l S /3' -j- and e, S 7 -|- e'l S y' -|- 



For the factors a, a' , ^, ^' corresponding to the equations most important 



in their influence upon the final determination of .r, y respectively, if we use num- 



bars chosen from a series for which ■ is only — as large as it is for the series (2), 



we shall have 



And if at the same time we omit altogether a certain number of the unfavorable 



equations by making in these instances a = 0, /3 = , that is, S a = — aw, 



S/3 = — b w, ov h ^^ — 1, we find 



° 631 



We shall therefore keep within the limits (28) and (29) as long as the coefficients in 

 the omitted equations satisfy the conditions 



^ j_ (?) 1 



(P)"^63l' (Q)^63l' 



The probable values of w — a'i, 1/ — y„ will not have been increased, and conse- 

 quently the solution may be accepted as equivalent to 11. 



A general method, III., of adjusting the degree of numerical accuracy which should 



be observed in the expression of the factors «, a' , ^, 13' , may be derived 



from the following considerations. 



In II. the adjustment is e\idently not so favorable as it might be, since the limit of 

 the intentional inaccuracies S a, S a , 8 /3, S /3' has been fixed by the relations 



8a = aio g, Sa' = a' w' g', 8 ^ — b w g, 8^' — b' w' g', 



g having the same average value whether aiv, hw be large or small ; thus the 



VOL. VI. NEW SERIES. 26 



