REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
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(V.) If the improved value of A, using certain auxiliaries, is A + a, and that of B, 
using some only of these, is B + 6, the m.p.e. of A + a and B+/3 is equal to that of 
A + a and B. 
(VI.) If the improved value of A, using P, Q, R, ... as auxiliaries, is 
A + pP + qQ + rR+ ... , the values of p, q, r, ... are given by a set of linear relations 
between p, q, r, ... and the m.pp.e. of A and P, Q, R, ... . [The equations are given 
by (III.), viz., since (A +pP + qQ+rR + ... ; P) = 0, &c., 
(A;P)+p{P; P) +g (Q ; P) +r (R ; P) + ... = 0, 
(A ; Q) +p (P ; Q) + q (Q ; Q) +r (R ; Q) + ... — 0, 
(A ; R)+p{P ; R)+q(Q ; R)+r(R ; R) + ... = 0, 
&c. 
It is assumed that the determinant 
(P ;P) (<? ; P) (R;P). . . 
(P ; Q) (Q ; Q) (R ; Q) ■ ■ ■ 
(P ; R) (Q ; R) (R ; R) . . . 
is not zero ; i.e. (see § 2 (iii.)) that there is a set conjugate to P, Q, R, ... .] 
(VII.) For any particular set of auxiliaries there is one and only one improved 
value of A. [This follows from the fact that the equations in (VI.), on the 
assumption there stated, have one and only one solution.] Hence we get the 
converse of (III.):—- 
(VIII.) If x is the sum of w and a l.c. of P, Q, R, ..., and, if the m.p.e. of x and 
each of P, Q, R, ... is zero, then x is the improved value of w, using P, Q, R, ... as 
auxiliaries. As a particular case :—- 
(IX.) If the m.p.e. of w and each of P, Q, R, ... is zero, the improved value of w, 
using P, Q, R, ...as auxiliaries, is the same as the original value. 
(X.) The improved value of P, using P, Q, R, ... as auxiliaries, is P —P - 0. 
[This follows either from (VIII.) or from taking A — P in the equations in (VI.).] 
Hence 
(XI.) If w is a l.c. of A, B, C, ... P, Q, R, ... , the improved value of w, using 
P, Q, R, ... as auxiliaries, is the same as that of the quantity obtained by adding to 
w any l.c. of P, Q, R, ... ; and, conversely :—- 
(XII.) If the improved values of w and of w', using in each case the same set of 
auxiliaries, are identical, then w and W either are identical or differ by a l.c. of the 
auxiliaries. 
(XIII.) If the improved values of A, B, C, ... , using in each case the same set of 
2 g 2 
