REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
205 
(iii.) The above relations can be expressed diagrammatically, thus :— 
• 
■ 
• 
- 
• 
• 
<5 3 
X 
X 
X 
X . 
, , 
b 
<52 
X 
X 
X 
X . 
• . 
b 
*1 
X 
X 
X 
X . 
. . 
b 
<5 0 
X 
X 
X 
X . 
• • 
O 
b 
u 0 
u x 
u 2 
u 3 . 
• • 
s* 
S* . 
The crosses represent the ( ) coefficients if they are the coefficients of the <fs in 
the us and of the y s in the <r’s, and the { } coefficients in the converse case. 
(iv.) Similarly, if we write (r = 0 , 1 , 2 , I ; t — 0 , 1 , 2 , ... 1) 
[rj — m.p.e. of y r and <r u .(22) 
then 
Vr — M <5 0 + [^l] <5i + [^ 2 ] 4 + • • • + [ r J .(23) 
o'* = [ffi] ^0+ [l<] u i + [2e] u 2 + ... + [/ t ] u t .(24) 
5. Sums as Conjugates of Differences. —The cases of importance are those in 
which the S’s are successive differences of the us. It will be found that in these 
cases the o-’s are l.cc. of successive sums of the ys. 
(i.) Let the <fs be the advancing differences of the us, i.e. 
S 0 — U 0 , rlj = Aw 0 , ... S r = A Uq, .... 
Then the diagram for the ( ) coefficients is 
• 
■ 
■ 
- 
<5.2 
0 
0 
0 
1 . . . 
CO 
b 
0 
0 
I 
3 . . . 
C^ 
b 
<5! 
0 
1 
2 
3 . . . 
b 
<5o 
1 
I 
1 
1 . . . 
b 
u 0 
u x 
u 2 
u 3 . . . 
