REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
211 
A, B, C, ... P, Q, JR, ... by S 0 , S u S 2 , ... <5) +1 , S j+2 , ... <1,. The order is quite arbitrary, so 
far as any general theorems are concerned ; but it will usually be convenient to place 
the auxiliaries last. If, for instance, we are using all but j +1 as auxiliaries, we 
denote those not so used by S 0 , S u S 2 , ... S j} and the auxiliaries by S j+1 , S j+2 , ... S t ; the 
improved values are then denoted by ( 
(ii.) We use the following notation :— 
(e f )j = improved value of cy, using S’s after Sj ; 
E ; = (ef = improved value of <5), using all subsequent <fs ; 
(\. g )j = m.p.e. of (e f )j and ( e g )j ; 
A . = = m.s.e. of E ; -. 
Where there is no doubt as to the <fs used as auxiliaries, (ej)j and {h./, g )j can be 
replaced by e f and \ f<g . 
(iii.) By (X.) of§ 6 — 
(e / ) j = 0if/>i;.(36) 
(vjj = 0 if/>y or g >j . 
(iv.) By (IY.)- 
(\, g )j = m.p.e. of (e f )j and S g 
= m.p.e. of Sj> and (e^ 
A . = m.p.e. of E; and Sj .(39) 
8 . Improved Values in terms of Conjugates. —In “Fitting” I have given some 
formulse for improved values in terms of sums. These may be regarded as derivable 
from a general theorem relating to the expression of improved values in terms of 
members of the conjugate set. The theorem is given by (XIX.) and (XX.) below; 
(XVIII.) is a particular case. 
(i.) Take any one of the <fs as S 0 . By (6)—- 
00 — *70,0^0 + >71.0^1+ ••• +>7z,o<^- 
Hence cr 0 /%, 0 differs from S 0 by a l.c. of the other cTs. Also the m.p.e. of <x 0 /%, 0 and 
each of these other S's is zero. It follows from (VIII.) of §6 that cr 0 /% i0 is the 
improved value of <5 0 , using the other S’s as auxiliaries. The m.s.e. of this improved 
value is %,o/(%,o ) 2 = lAo.o- Hence— 
(XVIII.) The improved value of any member of the original set, taking all the 
other members as auxiliaries, is the product of the corresponding member of the 
conjugate set by a constant; this constant being = the m.s.e. of the improved value. 
(ii.) The first part of the more general theorem is :—• 
(XIX.) The improved value of any l.c. of a set of quantities, using those after the 
first j +1 as auxiliaries, is a l.c. of the first j + 1 of the conjugate set. 
(37) 
(38) 
