232 
DR. W. F. SHEPPARD ON 
The formula which would have to be applied to the us in order to obtain this result 
can be constructed without difficulty. The important thing to notice is that, if we 
alter the differences of the us and then obtain the tv’s from the altered differences by 
summation, the resulting values must be such as can be legitimately substituted for 
the us. Suppose, for instance, that j = 2&+1, and that we use 2n+l us for each w. 
The formula for w will have to be of the form 
^0 > 
w {) — V A) + C- 2 k + 2^ + M(i "P C 2 £ +4 o + U(,-\- ... + C 2n (i' u 0 
and this will give 
^ k+i w h - r + s+c M+2 ^ +3 ^+c 3A+4 (^ +8 ^+...+c 2n ^ +2t+i ^. 
The problem of determining the c’s so that the m.s.e. of S 2k+1 w* shall be a minimum 
is the same as that of determining the coefficients in the improved value of for 
j = 4&+1 or 4& + 2, m being 2ti + 2ic-\-2 ; and the solution of this problem is given in 
§ 21. Thus, in terms of sums, (139) gives 
g — 2k +1 
t — ti *T k +1 
S 2k+1 wi 
^•2k + l,2g-lU ,7 ' 2gu i ) 
t = - (n+4+1) 
The X’s having been found, we shall then have, by summation, 
t = n+A+l 
t = — (n+*+l) 
' <7 = 24 + 1 
II 
df 
<>> 
- 2 X 2 , + , 2ff _iM^ +2t+ ffi 0 ; A k+2 u t 
The ratio of the m.s.e. of S 2k+1 w i to that of $ 2k+1 u i is given by (148). 
Appendix I. —The Correlation-Determinant. 
1. The m.p.e. of A and B being denoted by (A ; B ), let 
(A - A) (A; B) (A ; C) . . . 
(B ; A) [B ; B) (B ; C) . . . 
(C;A) (C;B) (C ; C) . . . 
We call this the correlation-determinant of A, B, C, .... 
'2. The elements of this determinant may be regarded as obtained as follows. We 
first take a representative collection of N A values of the error of A ; N A being usually 
indefinitely great. Then, for each of these values, we take a representative collection 
of N b values of the error of B ; the resulting N A collections will all be alike if the 
errors of A and of B are independent, but not if they are correlated. This gives 
N a N b combinations of an error of A and an error of B. For each of these we take 
