I 
OF THE COMPOSITIONS OF NUMEEIiS. 
875 
(2^ 
- (23 
+ (2^ 
KiK 2 
2)b, = % P, 
2)J.3=2P.„.-SP.,..S.. 
2)b,= 'i P........ - X P.,..,.S.. -I- X P.„. S,S.. 
(_).(2«_2);). = XP.,„,,^..-XP.„.......S..+ . . . + (-)"XP.,„S.., . . S, 
Now 
S P.1..S.. = X P.,..{S.. + S.. + . . . + S..) 
X P.,..S..S., = X P.,., (S..S.. + S.,S,. + . . . + S...,S..) 
with similar relations in the case of the other symmetric functions. 
47. Hence, by addition, 
(2=- 2)6i,-(28 -2)h3 + . . . +(-)-(2'- 2)6. 
= XP....(l-S,.)(l-S.,)...(t-S,.) 
+ X P,„.. (1 - S„) (1 - S.J . . . (1 - S,.) 
+ . . . 
+ ^ — S^„) 
I ^ Pk]K2 • • • 
(2^ -2)K- (2^ - 2) Zi, + ■ . . + (-)-‘ (2- - 2) h 
(1 _ SO (1 - So) ... (I - S„) 
P, 
and thence, 
= S 
KiK .2 
(I - S..) (1 - S.0 
+ ^ (I - s,o (1 - s,o (I - s,,) 
+ . .. 
p. 
Ki< 2 • • • Kl 
V- 
(1 — (1 — S^j ... (I - S,<,) 
P O.. 
(I - SO (I - so... (1 - S„) • 
5 T 2 
