122 ON THE DETERMINATION OF THE VALUE OF A CERTAIN DETERMINANT. [l 86 
gives rise to the terms 
( )°M XtS A x—2, S Hx—i8—2 ( X—-2, S H x _ 2g, 
(since \8~ — (it — 2s — l) 2 } H X _. 2S _. 2 = H X _. 2S ), or, what is the same thing, 
( ) 1 A i-y-] If x _2S ( yA X—-2' .S' H x —ng 
= ( ) \M X 'S— 1 A x _ 2<s _ 1 + A x—2 t s( H X —-2S> 
and moreover 
TJ X -4 contains the term (—) s A x _ i s H x _, M _ 4 , 
or, what is the same thing, (—) s A x _ i>s __ 2 IT x —is • 
Hence we must have 
A X g (A x _ 2 ' S + J\T X , ,5—1 H fl5 _2,8—3.) "4 (&’ 2) (it' 3) (/l it -f- 3) (/l it 4- 4) Ax—4's—2 = 0, 
where 
M x<s _1 = (# — 1) (ft — x + 2) + (it — 2) (w. — x + 3) — (it — 2s + l) 2 . 
This may be satisfied by assuming 
A x ,s = B X8 (ft— it + 1) (ft— it + 3)... (ft— it + 2s- 1); 
for then 
A x _ %8 = B x _ 28 (ft — it + 3)...(ft-it + 2s-l)(ft — it+2s+l), 
■Ax—2,8—1 =: B x —2,s—\(A *t H - 3) . . . (ft it + 2s — 1 ), 
(n — x 4- 3) (??• — it + 4) 4 s _ 2 (ft — it + 4) (ft — it + 3) ... (ft — it + 2s - 1), 
and consequently 
B x g ( ft - x + 1) 
— B x —2 t s (ft — x + 2s + 1) 
Bx—-2,S—1 1 
+ B x _ i8 _. 2 (.x - 2) (it - 3) (ft - it + 4) = 0; 
and if this equation be satisfied independently of n, we must have 
B x ,s B x _. 2 s (2it 3) 14—2,*—i -f- (it—2) (it — 3) B x _ i s _, 2 — 0, 
14,s - (2s + 1) 14-2,8 - {5it - 8 - (it - 2s + l) 2 } 14- 2 ,8-i + 4 (it - 2) (it - 3) 14_, M _ 2 = 0, 
and these are both satisfied by 
R _it.it — 1 ... it — 2s + 1 
*■* 2®.1.2.3...s ’ 
in fact, substituting this value and omitting the factor 
(it — 2) (it — 3)... (it — 2s + 1) 
2M.2.3...S 5