﻿196 ON THE USE OF EQUIVALENT FACTORS 



Let US suppose the mode of solution (II.) to be itself varied by changing the factors 

 a, a', &c., corres]3onding to the large and small terms, so that for the large terms, g, or 

 the probable value of — , ^-^, , for these particular terms, becomes 



aio a xo ^ 



g=H, 

 and for the small terms, 



g = h. 



We shall then have the probable values, 



For the large terms (8 P.) = IP (P), 

 For the small terms (5^,) = Ji^ (p). 



(B P) and (Bj)) becoming (8 Pi) and (Sj^J when g becomes H and h. 

 In order that the probable sum of the second member of the equation, 



P (x — X,) + P (i/ — J/,) + = f, S a + e'l S a' + 



should not be increased by the proposed changes of B a, we must have 



or, by (2b) and (27), 



H' (P) + h^ (P) < g' (P) + g"- (/>). 



"We shall assume, for the terms correspondmg to (j>), that the probable -salue of h is 



h = —1. 



This condition involves only small changes in the factors «, a , because, for the 



terras corresponding to (j)), a w being small, B a =^ a w h =^ — a lo, will also be small ; 

 we then have " 



H' {P)< g" (P) + ig' - 1) (P), 



or, since we can put (j~ — 1 = — 1 very nearly, g being small compared with unity, 

 we obtain 



(28.) ^» _ if^ > M , /f^ < ^ _ M , 



s -^ (P) ' ^ ^ (P) ' 



representing the condition to be observed in order that the second members of (24) 

 should not be increased by the changes made in the large and small values of a. 



This, it will be remembered, can be applied only Avhen the condition A = — 1 in- 

 volves only small changes in the factors a, a of the order of the mean value of 



B a for all the factors, y^ being necessarily a positise quantity, H must always be 

 less than g. 



