REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
227 
Substituting in (ill), and rearranging the terms, it will be found that the coefficient 
of A j u r is 
U+r, j) 1) F { - m +i+r+1, j+r+ 1 ; i,2j+r+2} 
= (/ + ?’,/) 2j + 1 ] ; 
so that 
(A) E, = 
Similarly from (112)—(115) 
(m—r— 1, /) 
r = m—j—1 
2 (/ + r, j) /VY AH/ - 
r = o [m,2j + \\ 
(116) 
(B) e 2 , = r= r (*»+*+*;, 24) (»+4 -r^4j r Ur 
v 7 2k (m, 4&+1] 
r = —n + k 
r = n—k 
(117) 
. (C) I M _, = 2 (ra + ^ + r, 2 k 1 ) (n + 4 r 1 . 24 1 ) ^_ v_ r _ >+ . ( 118 ) 
r = 0 
/to Tf> _r-nk ^ n + j c + r _i t 2k—l) (n + k-r -1 , 2&-1) 
V-L ') -*--'24—1 " / , / ,7 ® ^r+i> 
r = -»+* (m, 4&—1J 
r = n—k 
(E) I I( _i = 2 
r = 0 
{ft Tc v — 1 , 2 Jc — 2 ) {fi -\- Jc — v — 2 , 2 Jc — 2 ) ^ 24_2 
( m, 4& —3] 
. . . (119) 
{$ 2k - 2 u_ r + S 2k - 2 u r+ 1 ). . (120) 
The identity of (117)-(120) with (116), the us being altered as explained above, is 
easily verified. 
(v.) The formula of progression (77) takes simple forms if we attach the factors 
involving m to the <fs ; for m then disappears from the formula. 
(A) Writing 
A t = (m,t+ 1)E ( = (m, t + l) (A*v 0 ) t , 
so that, in effect, we take <5y to be (m,f+l) A /u 0 , (77) gives 
(A) (m,/+l) = s' YY-'lY 1 .( 121 > 
For / = 3, for instance, we should have 
(m, 1)% = A 0 
(m, 2) Av 0 = 
(m, 3) A\ = 
(m, 4) A 3 v 0 = 
~A 1 Ar\A 2 —\A 3 
A x —•§ A 2 +§ A 3 
A 2 —2A 9 
A 
where A 0 is the value of ( m , l) v 0 for j — 0, A 1 is the value of (m, 2) Av 0 for j = 1, 
and so on. This may be verified by 5 5 (i.)-(iii.) of “ Fitting.” 
(B) (C) Writing 
P2t = \A n > 2t + 1) E 2J , Q 2t _] EE (| ni, 21\ I 2t _!, 
