﻿IN THE METHOD OF LEAST SQUARES. 195 



Returning to the equations (10), 



P (.r — 1-,) + P' (y — y,) + = e, 8a + c', S «' + 



(24.) p/(^_,,)^Q (y_yO + = e,8/3+e'.8/3'+, 



we find, for the probable values of their second numbers, 



g /t \/w a a -\- w' a' a' -{- = g i^ VP , 



(25.) g,iViebb-\-w'b'b'-\- = gi^VQ, 



It is evident that the probable sum Cj S a + e', S «' + , being proportional to the 



square root of the sum of the squares of the individual terms, depends mainly upon the 



,— di 



large terms ; or, since Cj 3 a = Ci Vw -^ and e, Vw = At, this sum vdll be 





2 



+ 



If any two or more of the coefficients -^, as, for instance, ~^ and -7^, were eauaL 

 any small change increasing the former and diminishing the latter by equal amounts 

 would not alter the coefficient of a; but if —,-=- were much smaller than —A, we 

 should have, very nearly, 



WW w 



and a small change in r/=^ would affect the coefficient of /a by an amount insensible 

 compared with the effect of an equal change in ~7=^- 



Let (P) represent the sum of a certain number of the largest of the terms composing 



the series 



P ^ w aa -\- w' a' a' -\~ 



and (2)) the sum of a number of the smallest of the terms of the same series. Let also 

 (B P) be the sum of the terms — ^ corresponding to the series (P), and (Sj)) the sum of 



the terms — , corresponding to the series (p). 



Then we have the probable values 



For the large terms, (S P) = g-' (P), 

 For the small terms, {8p) =: g^ (p), 



g representing the general probable value of — for aU the terms, whether of large, 

 small, or medium value. 



