ON THE PROPERTIES OF A CERTAIN 
[2 
Substituting for the quantities involved in this expression, and putting, for simplicity 
2i + 2 — n = 2y, we have, without any further reduction, except that of arranging the 
factors of the different terms, and cancelling those which appear in the numerator 
and denominator of the same term, 
( -l)*fc x _ ( - l) g ~ x (1 -7) (2-7) ... (x-7) 
1.2 s 2 3S+1 .1.2...3.1.2... (s — x). 1.2 ... x 
multiplied by the series 
(« + S + 1) ... {% + S + X — 1) into 
. 7 x y (t±!) x ( x ~ !) 
1 x — 7 1.2 (x — 7) (x — 1 — 7) 
+ ... 
(x + 1) terms 
i + s) ... (i + 8 + x — 2) . 
into 
1-7 
Y 7 X (x- 1) 7(7+1) X (x-1) (x -2) 
+ 1 x — 7 1.2 (x — 7) (x — 1 — 7) 
/ i\r(^ + s —r+ 1) ... (1 + s + x-r- 1) . 
+ (-i) r - 7Ï H1 S into 
(l-7)(2-7) ... r-7) 
x terms 
x(x— 1) ... (x — r + 1) + y 
7 x(x-l)...(x -?•) 
x-7 
+ ...(x + r — 1) terms 
to r = X. 
Now it may be shown that 
1 
(1 -7) ( 2 -7) ••• (r~7) 
ix (x — 1) ... (x-r+lH^- 1 ^" ( X "~ r ) + &c. ...(x + 1 -r) termsl 
(x — 1) ... (r 4-1). x (x — 1) ... (x - r + 1) 
(1 -7) (2-7) ... (x-7) 
which reduces the expression for k x to the form 
( - l) 8+x 
(~l) s fc x _ 
1.2...S 2 2s+1 . 1.2 ... s . 1.2 ... (s — x) 
(f+s+l) ... (¿ + S + X —1) 
-j(i+s)... (¿ + s + x-2) 
+ X( l~2 1) (i + s-l)- (i + s + x- 3) 
+ &c. (x + 1) terms; 
from which it may be shown, that except for x = 0, k x = 0. 
The value x = 0, observing that the expression 
(i + s + 1) (i + s + 2)... (i + s — 1) 
represents , gives