﻿198 ON THE USE OF EQUIVALENT FACTORS 



largest inaccuracies are committed when aw,hiv are largest, that is, when the 



equation has most influence upon the final result for any particular indeterminate. 



In order to secure a more advantageous distribution, it will be necessary to recur 

 again to the equations (10). It appears that, for a given limit of inaccuracy in the 



expression of the factors, the probable values of x — x^, y — y^ will be least 



when the separate terms of the second members of these equations, irrespective of their 

 signs, are equal to each other, or 



Ci S a = e'l S a' = e, 8 /3 = e'l 8 /3' = 



Sa, B^ ought therefore to be inversely proportional to the probable errors of the 



primitive equations, or directly as the square roots of their weights. 

 We shall, then, define III. by the relations 



Sa = Ag Vw, 8 ^ = B g Vw, 



(30.) Ba' = Ag Vw' , 8 ^' = B ^ Vw' , 



A, B and g being constant quantities. 



To secure the equivalency of II. and III., the values of A, B must depend on 



the condition that the probable values of the second members of (10) should remain 

 unchanged, or 



g^'s/K' + K' + = gi^^P, gy^>/W-\-W + = gM VQ, 



A v^ = Vp, B Vrt = Vq, 



Hence it is easy to conclude, that, if we make in (30) 



A = mean value of a Vic, B = mean value of J \w. 



the means being in every instance taken without regard to signs, the probable values of 



X — x^, y — yy will be smaller in III. than in II., while III. in point of facility 



has a decided advantage over II. ; since by making a ^ 0, /8 = in all cases in 



which a Vw <^ kg, b \/w <.'Bg, , a considerable amount of unnecessary computa- 

 tion may often be avoided. 



The following will then be the limits of intentional numerical inaccuracies allowed 

 in the expression of the factors a, a , /3, /3' in the three methods. 



I. II. III. 



S«=0, 8^=0 8a=awg,Sfi=hwg S«=Ag^ Vw ,8/3 = B ^ Vw 



8 b' = 0, 8 ^' = ba' = a' w' g,h^' —I'w' g 8 «' = A g Vw', 8 j3' = B ^ Vw*' 



