REDUCTION{ OF ERROR BY LINEAR COMPOUNDING. 
203 
( b) If the m.p.e. of each pair of quantities of the original set is 0, but the m.ss.e. 
are not all 1, this is also the case for the conjugate set. The original set being 
A, B, C, ..., the quantities of the conjugate set are A/(A ; A), B/(B ; B), C/(C ; C ), ; 
and their m.ss.e. are l/(^4 ; A), 1 /(B ; B), l/(C; C), _ 
3. Relations between Original Set and Conjugate Set. —For expressing a member 
of either set in terms of the members of the other set, it is convenient to give them 
a linear order. We therefore denote the members of the original set by <5 0 , ... 
and those of the conjugate set by <r 0 , <r 2 , ... Also we write 
f r ,t = m.p.e. of S r and S t = f <>r ,.(2) 
>/ T , t = m.p.e. of <r r and <r t = r, Ur .(3) 
(i.) The condition of conjugacy is that (r = 0, 1,2,.../; t — 0, 1, 2, ... /) 
m.p.e. of S r and <r f = 0 (r ^ t) or 1 (r = t) .(4) 
(ii.) The expression for S r in terms of the <x’s is (cf. § 2 (iv.)) 
K = fr, O^O + fr. 1*D + fr, 2^2 + • • • + fr, l a f 
( 5 ) 
[For, if we write 
then (2) and (4) give 
S r = a 0 cr 0 + a^cr \ + agr^ +... + a t c r 
£ r t = m.p.e. of S t and a 0 <r 0 + a^x + • • • + a l <r l 
= «*•] 
(iii.) Similarly the expression for <r f in terms of the Ys is 
A + Vl, A 1 + %, A + • ■ • + *ll. A- 
( 6 ) 
(iv.) The relations between the f’s and the »/s are easily deduced from (5) and (6). 
If we write 
.( 7 ) 
z = 
fo.o 
fo, 1 
fo. 2 • • 
• fo.z 
fl,0 
fi.i 
fl, 2 • • 
• fl, l 
fz.O 
^2,2 • • 
• £>2,1 
fro 
fz.i 
fz, 2 • • 
. fz.z 
Z PtQ = cofactor of £ p>g in Z = Z QtP , 
(8) 
