of an Elliptic Transcendent Identity. 439 



/ d\ n ~ l 1 1.2...(n— 1) 



\ dn) , V* ( . ^\" 



= Tn.n- n .e & 

 = \/ ~e~ n e c-2. 



since, n being large, T(/i + 1) = \/2ft7r . »" . e~\ Thus we have 

 shown that 



e~~^ + e w/+e v <? J + . . . 



V 2ttV <fo/ L*/» ^n J 



so that it only remains to find the value of 



/ d\ n - } e~ b * /r 

 \ fifo/ Vn 





viz. of 



(rs) v 2 - &+ r^^-iT2T3 ,l+ ---) 



_ l-3-.(2n-3) ^ l f 1 _JT_JL_ 



2"" 1 L 1.2 2/1-3 



£ 4 1.3.n g I 



+ 1 . 2 . 3 . 4 (2n-3)(2w-5) + * ' ' J 



__ r(2n + l) »/ y y _ % 



~2 2 »/iI> + l) L 4^1.2.4 2 J 



2 2 »w«+ie-V27r 



/o _** 



V ft 



whence 



e ^2 + e v c * -fe V c ) + . . . 



= _ (1 + 2e~ c2 cos 2^ + e" 4c2 cos 4a? -f . . . )„ 



\ /r TT 



which, when transformed by writing x for- andafor— ,becomes(l) 1 . 

 It is interesting to observe in the above investigation how the 



