OF THE COMPOSITION'S OF NUMBERS. 
873 
Since M - N = ( 2 ^ -2)h,- (2^ - 2 ) + . . + {-Y ( 2 - - 2)h,, 
N _ ]\I 
D "" D 
]\I - NT 
D 
V 1 
1 _ (2^ - 2) 5^ - (2^ - 2) + . . ■ + (-Y (2« - 2) h 
s/ 
VC + f^ 
(22 
(1 - Si) (1 - S 2 )... (1 - S„) 
+(_)«(2»-2)&„. 
(1 - Si) (1 - S,) ... (1 - S„) 
the second member of the right-hand side of this identity has now to be transformed 
into a series of partial fractions of the same form as those which arise from the 
product 
43. Let /Cl, /C 3 , . . , Kt be any t different numbers selected from the first n integers 
1 , 2 ,... n, arranged in ascending order of magnitude. 
Let 
= 2s^a^S.OL^ 
<\<2 ^2 ^^2 
f>(*) _ Q _|_ *150) Q I Q 
-^/CxK2'f3 n ^/C2 I -^Ki/C2 ^^3 
B (l) _ V Q Q 
the summation being for the terms obtained from the expressions that it 
is possible to construct from the t integers Ki, /Co, . . . /c/. 
Further, let 
S; + 0 . . • S; 
Bl'l .. = S 
'K1K2 . . , Kt 
<^<2 . . . Kj + l + i 
the summation being for the ^ terms obtained from the (^j ^expressions 
B^i/c,... * 1 + 1 ^^ that it is possible to construct from the t integers k^, /c^, . . . k^. 
44. The following lemma is required :— 
“ Lemma. 
V S = 
^ ^K\K2 . . . K(-i 
K 1 K 2 • • • Kl^ 
= i bB«> 
K^K 2 • » . Kl ^ 
5 ' T> 0 ) Q 
“* ■’^*1*2 • • • l ’ 
MDCCCXCIII.—A. 
5 T 
t -j-1 
B 
O') 
’K1<2 • < • Kt 
