REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
201 
values. In doing this I found that the whole of the work could be very much 
simplified by using certain general theorems, which applied not only to the special 
case of the standard system but also to the general case, and even to a still more 
general problem in which, in the one aspect, the reduction of error is effected by 
means of quantities which are not necessarily a set of differences, or in which, in the 
other aspect, U r is not necessarily a polynomial in r of degree^’ but is a linear com¬ 
pound, with coefficients to be determined, of any^’+l functions of r ; and the present 
paper is mainly concerned with these general theorems, so that to a certain extent it 
supersedes the previous papers. 
The abbreviations l.c., m.s.e., m.p.e., are used for linear compound, mean square of 
error, mean product of errors. The mean square of error of A is denoted by (A ; A), 
and the mean product of errors of A and B by (A ; B) or (B ; A). Other special 
notations used in the paper are the same as in the three papers mentioned at the 
beginning of this section, or are explained in §§ 3, 5 (iii.), 7, 17, and 20. 
Conjugate Sets. 
2. Conjugate Set. —(i.) Let A, B , C, D, ... be a set of quantities, not necessarily all 
of the same kind, containing coexistent errors which are either independent or 
correlated in any way. For the purpose of the following investigations it is 
convenient to consider, in connexion with these quantities, another set of quantities, 
G, H, J, K, ... , equal to them in number and connected with them by the conditions 
that (l) each quantity of the second set is a l.c. of those of the first set, and (2) the 
m.p.e. of corresponding members of the two sets is 1 and that of members which do 
not correspond is 0. If we replace the quantities of the two sets by A 0 , A 1} A 2 , ... , 
and 6r 0 , Gi, G 2 , ... , we can express this latter condition by saying that m.p.e. of G r 
and A s = 0 (s r) or 1 (s = r). The second set of quantities is said to be conjugate 
to the first. 
(ii.) Let the member of the second set which corresponds to C of the fust set 
be J. To determine J, let us write 
J — aA + bB + cC+ dD +_ 
Then, denoting the m.p.e. of A and B by (A ; B), condition (2) gives 
(A ; A) a + (A ; B) b + (A ; C) c + (A ; D) d+... = 0, 
(B ; A) a + (B ; B) b + (B ; C) c + (B ; D) d+ ... =0, 
(C ; A)a + (C ; B)b + {C ; C)c + {C; D) d+... = 1, 
(D ; A) a + (D ; B)b + {D ; C)c + (D ; D)d+.. = 0, 
&c. 
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