DEDUCTION OF ERROR BY LINEAR COMPOUNDING. 
225 
have the values of the e’s in terms of the <x’s ; and also, by § 10 (iii.), the values in 
terms of the As (which are identical with the y s). This completes the investigation 
of the improved values; but we also want to see the extent of the improvement. A 
further section therefore gives the ratio of the m.s.e. of the improved value to the 
m.s.e. of the original value, in a form con venient for calculation. 
(x.) The m.p.e. of A hi r and A f u r+t is ( —)‘ (2 f,f+t). Hence, in finding the general 
solution of our problem for the case of a self-conjugate set, we are also finding it for 
the case of a set in which the m.p.e. of u r and u s is of the form ( —) sr (2 f,f+s—r), f 
being some positive integer ; for we can treat these As as the f th differences of 
members of a self-conjugate set. 
21. Improved Values. —-(i.) Adapting § 11 (iii.) to the case in which the y s are 
identical with the u’s, we see that, if w is any l.c. of the As, its improved value, 
using differences of order exceeding j, is of the form lLp r u r , where p r is a polynomial 
of degree j in r. The improved values of the As and their differences are obtained 
from this by means of certain formulae, given in §§ 6 and 7 (iv.) of “ Factorial 
Moments,” for the expression of lLp r u r in terms of differences. The results are given 
in (12), (18), (19), (26), and (25) of “ Fitting.” From the first of these, replacing 
m+ 1 by m, we have (f = 0, 1, 2, ...j) 
(A) A\ =7f A/) U-sJ-f) (Yftl'j'SfeA 1 ) A ’ M » ; 
( 101 ) 
and from the other four, altering k to k —1 in the last, we have [f— 0, 1, 2, ... k in 
(102) and 1, 2, ... k in (103)-(105)) 
(B) = T (t+ ., t +f) (k-s, k- f ) j; gg^f;ffiij • (w») 
= • (102i) 
(O) m<S + S 1 > k+ J S ’ L ”0(2£ + 2s-l, 2i-l)(« + i2/] M ° 
... (103) 
s-h 2/'+l {2h, 2/) pm, 2s] t 
s = fe+l S — f2k+l |{2 k, 2s} (£ra, 2/] 
m ^/-i M o+ (_)^-/ 2 pA 'J . • • (103a) 
(D) 
(k + s—i k + f _1) (k _s k f ) ^ ^ 1 ) 
A 1 ' J U ’ J> (2k+2s-l, 2k-l)[n,2f) * 
. . . (104) 
- W-IUL+(-Y-t *~ k 2 f +1 2 -^ 2s > S--'u. (104O 
_<i “ i+( ’ ,? 1 us-f2k+l {2k,2s}[hm,2f) d ' ’ ' ,I04A) 
2 i 2 
