210 
DR. W. F. SHEPPARD ON 
auxiliaries, are A + a, B + ft, C + y, ... , the improved value of any l.c. of A, B, C, ... , 
using these auxiliaries, the same l.c. of A + a, B + /3, C + y, ... . [Let the l.c. of 
A, B, C, ... be + bB + cC +... . We want to prove that x = a(A + a) + 
b (B + ft) +c (C+y) +... is its improved value. We can do this in either of two 
ways : 
(i.) By (III.), the m.p.e. of x and each of the auxiliaries P, Q, B, ... is zero ; and x 
differs from w by a l.c. of P, Q, R, .... Hence, by (VIII.), x is the improved value 
of w, using P, Q, R, ... as auxiliaries. 
(ii.) A more direct proof follows from the linearity of the equations mentioned in 
(VI.). It is not necessary to set out the proof here.] 
(XIV.) If A, B,... C, D, ... P, Q, ... B, S, ...fall into two classes A, B, ... P, Q, ... 
and C, D, ... , It, S, ... , such that the m.p.e. of each member of the one class and 
each member of the other class is zero, then the improved value of a l.c. of any of the 
members, using P, Q, ... It, S, ... as auxiliaries , is to be found by talcing the two 
classes separately, i.e., by using P, Q, ... as auxiliaries for the terms in A, B, ... P, Q, ..., 
and R, S, ... as auxiliaries for the terms in C, D, ... R, S, ... . [For the m.s.e. of 
aA + bB + ... + cC + dD + ... + pP + qQ + ... + rR + sS + ... is the sum of those of 
aA + bB+ ... +pP + qQ +... and cC+dD +... + rR + sS +... , since the m.p.e. of these 
latter is zero ; we cannot reduce the m.s.e. of the first of them by adding terms in 
R, S, ... , or that of the second of them by adding terms in P, Q, ... ; and the result 
is therefore the same as if we considered them separately.] 
(XV.) If the improved value of w, using P, Q, R, ... as auxiliaries, is 
x = w + pP + terms in Q, R, ..., this is also the improved value of w + pP, using 
Q, R, ... as auxiliaries. [For x differs from w+pP by terms in Q, R, ..., and the 
m.p.e. of x and each of Q, R, ... is zero.] This can he stated more generally as 
follows:— 
(XVI.) If the improved value of A, using a set of auxiliaries S, is A + a, and if 
we divide S into two sets, Sj and S 2 , and the corresponding parts of a are a 1 and a 3 , 
then A + a is the improved value of A + a,, using S 2 as auxiliaries. [We may take 
... P, Q to be S u and R, ... to be S 2 . The theorem states that, if the improved value 
of A, using ... P, Q, R, ..., is A + ... +pP + qQ + rR+ ..., this is also the improved 
value of A + ... +pP + qQ, using R, _] 
(XVII.) The following corollaries of (III.) may he noticed, though we shall not 
require them. If the improved values of A and of B, using in each case the same 
set of auxiliaries, are A + a and B + /3, then 
(l ) (A + a ; A + a) = (A ; A) — (a ; a) 
and 
(2) (A+a; B + ft) = (A ; B) — (a; ft). 
7. Notation : and Particular Values. —(i.) It will now be convenient to adopt a 
linear arrangement of the quantities we are dealing with, and we therefore replace 
