﻿190 ON THE USE OF EQUIVALENT FACTORS 



tain the ratio — , we will compare the probable values of Xq — cc and x — x\ , having, as 

 above, 



n" Probable value of ix — Xi) 



(12.) — = • 



* Probable value of {xq — x) 



Xo — X and x — Xy must be derived from a solution of equations (10) and (11), but 

 since (II.) differs from (I.) by small variations only, we have, very nearly, 



(13.) w e," + w' e'c + = w e^ + w' e'^ + 



For the second member of (13) is a minimum relatively to the mode of solution, and, 

 as has already been shown, (1), it differs from the first member by small terms of the 

 second order only, those of the first order vanishing with the first differential coefficient 



oi. SI ^= w e- -\- w e'~ -\- 



If, then, fiQ, /Li, and /i, represent the probable values of Cq, e, and e, corresponding 

 to the unit of weight of the equations (7), we may assume, for the purpose of deter- 

 mining X — x\r that /i — /x, is a small quantity compared Avith /i, since we have 



y? 7c e' + «?' e'= + , 



/ii 10 e, -|- vr e'l -\- 



Moreover, in the absence of exact knowledge of the magnitude of the errors of Bq, 



c'q, , it is necessaiy to admit that they are best represented by the errors e, e, ; 



hence we have — = 1, and consequently - = 1, very nearly. 



The conditions of the solution (II.) give for the probable value of either of the ratios 



Because a being by (II.) nearly proportional to a w, the probable value of S a will also 

 be proportional to a iv ; and a similar remark applies equally to S /8, S y, &c. 



The probable values of the second numbers of (10) and (11) are then, respectively,- 



8 a e, + 8 a' e'l -|- = g i^i \/7 ^ a to e^ -\- a' lo' e'o -{- = /lo \/P, 



8 /3 e, + 8 ^' e', + = g i^i V Q ' & Jo e„ + J' jo' e'o + = /^o \/Q, 



Hence, in consequence of the identity of the coefficients P, P' , Q, Q' , &c. in 



the two systems (10) and (11), we obtain 



