REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
207 
then 
A = 
r = ?i 
2 
r = — ?i 
Id > 2/<) y n+r , 
(27) 
°"2A-1 
2 (r, 2/i— l] ?/„ + 
(6) If the w’s are ti,„ rq. w 2 , ... w 2 „_ 1} so that 
(28) 
S n = 
then 
n-& ^1 — 
S 2 = uS 2 u n _ h S 3 = <fu n _ h ... , 
r = n 
°2A-1 — 2 
[r~h 2h-l) y n+r _ 1 . 
• • (29) 
r = - ra+1 
r =n 
cr 2h-2 = 2 
(r-h 2 /^— 2 ] y n+r _i . 
. . (30) 
r= —n+1 
(iii.) The values given by (25)-(30) may be expressed in terms of successive sums 
by the formulae given in “ Factorial Moments.” The notation, however, can be 
simplified. Suppose that we have a set of quantities ... F 0 , F 1} F 2 , ... corresponding 
to values ... 0, 1, 2, ... of some variable, and that we form the table of successive 
differences (and also, if we like, of successive sums) of the F’s. Then the Lagrangian 
formula for F e in terms of F p , F p+1 , ... F p+t , which can be expressed in a good many 
different ways, may be regarded as the formula for it in terms of the whole 
(unlimited) set of differences (and sums) which form a triangle with its apex 
at A t F p \ and we can denote it by L{F 0 ;A t F p }. With this notation, the above 
results may be written 
(25) 
■v=u{(-) / 2' t, »/i2».}];:f 1 . ■ • • 
.... (31) 
(27) 
C 2 h = [L {p.F h+v y n ; I • • • ■ 
.... (32) 
(28) 
rak-i = [L {—v h y n ; <ry n +t}J t Z n - r Lv • • • • 
.... (33) 
(29) 
r t r 2 h . \y = ra+$ 
°2A- 1 — L-L\—yc y n -i '■> cyn + t-lj\= -n+*» 
.... (31) 
(30) 
°'2A-2 = [L {o- 2A_ V»-i; ^y n +t-i}J t Z n -n+r ■ ■ ■ 
.... (35) 
The L is 
distributive as regards the first member inside the { } ; 
e.g., in the case 
of' (31). 
A<r a + Br, = [L ; Si/,}] 
t = 1-1-1 
t = 0 ‘ 
(iv.) More generally, suppose that the <fs are the successive differences of the u’s 
according to any system of differences ; by which we mean that S s is either a definite 
difference of the its of order s (the us themselves being differences of order 0) or a 
l.c. of such differences. Then (r t ) of (14) is a polynomial in r of degree t, and <r t is 
VOL. CCXXI.-A. 2 G 
