212 
DR. W. F. SHEPPARD ON 
For, if' w is a l.c. of S 0 , S u S 2 , ... we can (see §2 (v.)) express it as a l.c. of 
( 7 0 , o-j, <r 2 , . • • <Tj, $ J+1 , S j+2 , ... 4 Let the result be (2) + (A), where (2) is a l.c. 
of o- 0 , o- 2 , ... o-j, and (A) is a l.c. of S J+1 , § j+2 , ...§, Then (2) differs from w by a 
l.c. of these latter <fs, and the m.p.e. of (2) and each of these <fs is 0 ; hence, by 
lYTII ), (2) is the improved value of w , using these d’s as auxiliaries. 
(iii.) Further— 
(XX.) The coefficients of the <r's in the improved value of the l.c. are the m.pp.e. of 
this improved value and the corresponding 8 s. 
For, if the improved value of w is x , and we write 
x = b l) o- IJ + b 1 o- 1 + b 2 o- 2 + ... + bjo-j, 
then, by the condition of conjugacy of the o-’s and the <fs, 
m.p.e. of x and f = b f . 
(iv.) This would give us a solution of the problem of finding the improved value, 
if we could find the m.pp.e. Ordinarily, w is or can be expressed in terms of the cfs, 
and we do not find its improved value independently, but deduce it from those of the 
s up to Sj. The improved value of S A is, by (iii), 
( € h)j = (\,h)j<ro + {hi,h)j<ri+ .( 4 °) 
and the m.pp.e. that we really require are therefore the values of (f./f- With 
regard to this, see §9. 
(v.) As the converse of (XIX.) it may be noted that— 
(XXI.) A quantity of the conjugate set, or a l.c. of such quantities, cannot be 
improved by means of the non-corresponding quantities of the original set; e.g., 
a l.c. of <r 3 and <r 4 cannot be improved by using the d’s, other than S 3 and c> 4 , as 
auxiliaries. This follows from (IX.) of §6, since the m.p.e. of <^ r and o- s is 0 
unless r = s. 
(vi.) If, as in § 5, there are related conjugate sets of «’s and y s, and the cTs are the 
differences of the us, it follows from § 5 (v.) that (2) in (ii.) above is a l.c. of the 
moments of the y s up to the j th . (XIX.) is therefore a generalisation of the 
theorem, for a self-conjugate set, that the improved values are l.cc. of the moments ; 
and, in fact, it explains the appearance of the moments in this connexion. 
9. Mean Products of Error of Improved Values.- —(i.) We have found, in § 8 (iv.), 
that 
( e h)j “ (\h)j + (X,/,)j Ti + • • ■ + fj.h)j a j- 
To obtain the A’s, we introduce the condition that this shall differ from by a l.c. 
of <fs after <5). 
