874 
MAJOR P. A. MACMAHO^T ON THE THEORY 
the summations being subject to the same conditions as before. The truth of these 
relations is obvious from the elementary theory of permutations.” 
45. Consider now the series of functions— 
p _p(i) 
p _ p(l) _ p(2) 
p _ p(l) _ p(2) I p(3) 
p _pu) _p(2) 1 I /_ 
K1K2 • • • K( . Kl ■^KiK 2 • > • <t \ • * • 1 \ / K( 
From the foregoing lemma are derived the relations— 
S p,,.. S... . 
2 P.,,.A- 
2P S 
Q _ p(’) 
• ^Kl •••'<«? 
.. s = 
t 
0 
\ 1 
t - 2 
9 
B 
(1) 
K1K2 . . . Kt 
-B 
B 
(1) 
K1K2 . • . Kt 
(2) 
K1K2 • 
t ■ 
kO 
(3) 
KlKa , 
2P 
IClKS . 
s 
Ks+i 
s = 
B, 
( 2 ) 
K1K2 
+ •■■ + (-)' B 
(s-l) 
. Ki* 
Hence, by addition with alternate signs, 
/ — V B — P —YP S-I-'^'P SS — 
\ / . Kt K1K2 . . , Kt K1K2 . . . Kf^i *^Kt ^ KiKo . . • ’ * * 
+ (-)'2P..,.S..S.....S., 
Here the summations are in respect of the t numbers k^, k^, . . . K(. 
46. A similar identity may be established for every selection of t integers out of 
the numbers 1, 2, 3, ... n. 
Summing all these identities, and remembering that 
2B.,.,.,.„('-» = (2'-2)6,, 
we obtain 
(_ y (2' _ 2) 6, = 2 P,,..2 P.„ S., + 2 P..,. S, . . . S,,, 
or, attributing to t all integer values from 2 to n inclusive, we have the series of 
relations :— 
