2*22 
DR. W. F. SHEPPARD ON 
If we can obtain the 0 ’s and the A’s in a simple form, we thus have a workable 
formula for calculating the A’s, and thence, by (40), for determining the e’s in terms 
of the cr’s. 
(ii.) From (80), using (II.) of § 6 , we get the m.ss.e. and m.pp.e. of the improved 
values of any l.cc. of the S’ s. Let 
w — b^S (j -\-b x Si + ... + b t Si, w' — c 0 S 0 + Cjflj + + 
and let the improved values of w and w', using S’s after S j} be x and x'. Then 
t =j 
m.p.e. of x and x' = 2 (Mo.* + Mi.t + • ••) (c o 0 o ,« + c 1 0 1 , t + ...)\ 
( = 0 
—■' (^o^o,«"t&i0i,j+ ...+6j0 i t ) (c o 0 Oif + c 1 0 lit + ... + c t 6 t t ) A t , (8l) 
t = o 
m.s.e. of a: = 2 (6 o 0 o , t + Mi,t+ ••• +b t 0 tJ ) 2 A t .(82) 
t = 0 
(iii.) We have assumed that the 0 ’s are known. If they are not known directly, 
but the values of (Ay it ) t are known, then, by (79), 
fy.t — .(83) 
Substituting in (80), 
(V.?)i = ^ {\,t)t/\‘ .(84) 
t =/. <7 
Also (77) is replaced by 
(<A = 2 J (X / .,),E,/A l .. (85) 
t=.f 
Application to Self-Conjugate Set. 
20. Preliminary. —(i.) We have now to apply the preceding results to the case in 
which the us are a self-conjugate set, so that (u r ; u s ) = 0 (r^s), (u r ; u r ) = 1, y r = u r . 
We take the d’s to be successive differences of the us, commencing with a difference 
of order 0. The <fs to be used as auxiliaries will be those following Sj ; the ( ) 7 
will usually be omitted. We shall take the number of w’s or of d’s to be m, so that 
m — l + 1. 
(ii.) There wdl be three cases to be considered; advancing differences, and central 
differences for rn odd and for m even. For advancing differences the us will be 
taken to be u Q , u 1 ,... u m-1 . For central differences we shall write m = 2w + l or 
m = 2 n ; and the us will be u_ n , u_ n+1 , ... u„ and u_ n+1 , u_ n+2 , ...u n respectively. We 
