REDUCTION OF ERROR BY LINEAR COMPOUNDING. 215 
Hence, by (IV.) and (49) or (49a), 
p r — m.p.e. of a; and v r 
= ( r o) V.+ ( r i) V + ( r 2 ) X 2 + ... +(r z ) X;.(53) 
— ( r o) V + ( ? ’i) V+ (^2) V + • • • + ( r j) .(53 a) 
Thus the jo’s are related to the A’s in the same way that the Vs are related to the e’s, 
or the Vs to the d’s. 
(iv.) Similarly, if 
x = q 0 u 0 + q 1 u 1 +q 2 u 2 + ... +q t u h 
the g’s are related to the X’s in the same way that the z s are related to the e’s, or the 
ys to the d’s. 
11 . Special Case of Differences .—The important practical case is that in which the 
d’s are successive differences of the Vs, in the general sense explained in §5 (iv.). If 
the differences of order exceeding j are negligible, we can use them as auxiliaries for 
improving the Vs or the d’s or any l.c. of the Vs or the d’s. 
(i.) Since the Vs are successive differences of the Vs, ( r t ) is (§5 (iv.)) a polynomial 
in r of degree t. 
(ii.) By § 10 (i.) the e’s are the differences of the Vs according to the same system ; 
and v r is a polynomial of degree j in r, the differences of the Vs of order exceeding j 
being zero. 
(iii.) With the notation of §10 (iii.), the X’s are the differences of the p s according 
to the same system; and p r is a polynomial of degree j in r, the differences of the 
p’s of order exceeding j being zero. 
(iv.) If we form the differences of the Vs in the usual way, there will be l differences 
of order 1 , l—l of order 2, and so on. The l —j + 1 of order j , namely A% 0 , A j u u ... A j Ui_ jy 
will differ from one another by l.cc. of the differences of higher order ; and therefore, 
by (XI.), they will have the same improved value. If we denote this by E, then, 
if iv = pA ] u 0 + qA j u 1 + rA j u 2 + ..., the improved value of w is (p + q + r + ...)E. 
Relation of “Reduction of Error” to “Fitting” (of a Polynomial). 
12. Standard System .—In the case of a standard system, the process of reduction 
of error and the process of fitting a polynomial (by least squares or by moments) give 
the same result. The following is a proof of this, not involving the properties of 
conjugate sets. The observed values are taken to be u 0 , U\, u 2 ,... u t ; and 2 denotes 
t 
summation for t — 0, 1, 2, ... I. 
(i.) If the polynomial which we are fitting to the Vs is 
v q = a 0 + a 1 q + a. 2 q 2 + ... +aff, 
2 H 
( 54 ) 
VOL. CCXXI.—A. 
