REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
213 
(ii.) Substituting for <r 0 , <r,, <t 2 , ... from (6), this condition gives 
(\h)j ('lo. 0^0 + >1l, 0^1 + ... + H] U 0 S t ) 
+ ( A, h)j (%, 1 A + >71. A + ... + rjI' iSf) 
+ (\k\ j (%. 2 A + *11, 2 ( \ + • • • + Vl, 2 &i) 
+ . . . 
+ (\k)j (%Jo + mJi + ••• +>n.j$i) 
= $h+ terms in S j+1 , S j+2 , ... S t . 
Equating the coefficients of S 0 , S lt S 2 , ... S p we find that (/ = 0, 1, 2, ...j) 
if,o(\h)j + *if.i (\h)j+ ••• + r if,j (\h)j — 0 (/ ^ h) or 1 (/= h). . . . (41) 
Let us write 
H j = »/ 0 ,o %,1 %,2 • • • >h,j [ ..(42) 
*71,0 *7i,i J 7i. 2 • • • *h,j j 
%, 0 *12,1 *h,2 • ■ • *l2,j 
. 
J 7;,U *lj, 1 *lj,2 ■ ■ • fyj 
H ? , M - = cofactor of ^ in H, = H /it?; , 
Then 
% oH 0 , htj + r, ft jHi , htj h ,j = 0 (/ ^ h ) or H ; (/ = h) ; 
and therefore, by (41), 
(V*)j = Hj.j.j/H,. 
(43) 
(44) 
(45) 
Substituting in (40), we obtain (e h )j in terms of the o-’s. 
For the particular case of g = h — j, (\, h )j becomes A jt and H g htj becomes H 7 _j ; 
so that 
A, = rr^/FI,.(46) 
(iii.) As an example, suppose that we have several independent observations, of 
unequal accuracy, of a single quantity U, and that we wish to obtain a suitably 
weighted mean, which may be regarded as the improved value of any one of the 
observations. Let the observed values be u 0 , u u u 2 , ... , and their m.ss.e. 
a 2 , &i 2 , « 2 2 , ••• ; the m.pp.e. being 0 , since the observations are independent. We 
take S 0 to be one of the us, and <$ 1} S 2 , ... to be its successive differences. Then j = 0 , 
since the true values of the first and later differences are all 0 . Hence, by § 8 (i.), 
the improved value is <x 0 /(m.s.e. of <r 0 ). But, by § 5 (i.), <r u = 2 y ; and, by § 2 (vi.) ( 6 ), 
y r = u r /a 2 , so that m.s.e. of <t 0 = 21 /a 2 . Hence the improved value is 
2 (u/a 2 )/2 (l /a 2 ). 
