846 
MAJOR P. A. MACMAHON OR’ THE THEORY 
Dl — 5 
1^3 = ^ {(^h + l)j 
Dg = — (c?i + 1) {d^ + 2), 
O ; 
, di ((ii + 1). . . (c?i + ^ — 1), 
H' ■ 
the products on the right denoting successive operations."^ 
To evaluate D^H’’ as au algebraic function of H, it is necessary to operate 
successively with cZj, + 1,. . . c/^ + p. — 1 and to divide the result by p ! 
We have 
= rW-^ + rW, 
d,h-' = Qh-= + (’Jh- 
Suppose that generally 
\ 
+ s 
Operation with 
P + 
, I + p) 
gives 
3 = M + 1 /r -I- c _ 1 
s = 0 \ S 
p + 1 — s 
JJv - (;a + 1) + i 
SO that the assumed law is establislied inductively. 
* The truth of this law is seen by comparison with the corresponding algebraic relations. 
Denoting by • the sums of powers of a^, a„i • 
1 — a^x + aM" — = exp. — (sjce + isja;- + + ...), 
and putting 
l — a^x -\- a.TX^ — . . . = (1 -f 
Hence 
0^1 = _ Sj (i’l + 1) . . . (i’j + — 1), 
/I • 
from which we pass to the operator relation 
“ “1 ‘h (‘h + 1) • • • (dj + — 1). 
Observe also the inlation 
1 
