REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
219 
(v.) The ordinary method of least squares would consist of making 2 (v s —u s ) 2 a 
ft 
minimum, and would lead to equations 
2(s/) (v s -u s ) = 0, 
S 
which would not give the most probable values of the e’s. 
14. Fitting by Moments. —(i.) The ordinary method of moments, adapted to the 
case in which the S’s are not necessarily the successive differences of the u s, would 
consist in equating the values of 2 (sf v s and 2 (s f ) u s . This, as will be seen from 
8 S 
§13 (v.), would not give the most probable values of the e’s. 
(ii. ) In order to obtain the most probable values of the e’s by equating moments of 
the v’s and of the us, we must write (say) 
Mf — 2 [s^] u s , .(71) 
S 
and define the f th moment of the us as being M f or a definite l.c. of Mf, M f _ x , 
Mf_ 2 , ...M 0 . But the coefficient of u s in Mf would then not be given definitely by 
the relations between the us and the <fs, but would depend also on the law of 
correlation of errors of the its. We see, however, from § 13 (iii.), that we have also 
M f =X{r,)y r .(72) 
r 
and that we get the same result by equating moments, defined in this way, of the y s 
and the z’s. In the ordinary case in which the S’s are successive differences of the 
us, the coefficients of the y s in (72) are binomial coefficients, and the ordinary 
moments fall within the definition given above. It follows that in fitting a 
polynomial to a set of quantities (not being a self-conjugate set) by the method of 
moments, the moments which ought to be equated are not those of the quantities 
themselves and their assumed values, but those of the conjugate set of the former and 
the corresponding l.cc. of the latter. 
15. Reduction of Error. —If we improve the S’s or the ffis by means of the S’s 
after <f the improved values of these latter are zero, and those of the S’s up to j are 
obtainable from (XXI.) of § 8 , which states that the improved values of the a-’s from 
<x 0 to o-j are equal to the original values. Using (6), this gives (,/ = 0 , 1 , 2 , ... j) 
x rjf t (e t )j = 2 rjf t (\ .(f^) 
t =0 i=0 
Comparing this with (70), we see that the e’s given by this process are the same as 
those given by the process of fitting the expression in (57). 
16. Difference of the Two Processes. —Although the two processes lead to the 
