57] 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
365 
A = - 12 (n - 1) (n - 9) (47n 3 - 355n 2 + 3188n + 31500) a 
+ 20 (n — 1) (n — 9) (n — 25) (n — 49) (n + 18) a 3 
+ (n — 1) (n — 9) (ft — 25) (n — 49) (n — 81) a 5 , 
A = - 24 (n - 1) ( w - 9) ( 23?i 4 + 2375w 8 - 14638?2 2 + 116100n + 693000) 
-12 (to - 1) (n - 9) (493?i 4 - 8882n 3 + 70317w 2 - 361641n - 7276500) a 2 
+ 30 (n — 1) (n — 9) (n — 25) (n — 49) (n — 81) (n + 22) a 4 
+ (n — 1 )(n— 9) (n — 25) (n — 36) (n — 49) (n — 81) {n — 121) a 6 , 
&c. 
And, in general, 
A = (n — 1) (n — 9)... [n — (2r — l) 2 } a r 
+ r (r — 1) (n — 1) (n — 9) ... {n — (2r — 3) 2 } (n + 4r — 2) a r_2 , 
&c. 
(where however the next term does not contain the factor (n — 1) (n — 9) ... [n — (2r- 5) 2 }). 
In the case when n = r 2 , then in order that the constant term may reduce itself 
to unity, we must assume 
M = (-) i(, '~ 1) v ; 
this is evident from what has preceded.