MULTTPLTCATTOX OF ELLIPTIC FUXC'J'IONS. J 79 



whicli nuTecs \\\\\\ flie form of -7-- o-iven l)y R. iro|)i)0.* 



Put u = y'\ then —. — = /^. //'"''. where />,. denotes tlie continued 

 (///' J ' ■ 



product of /• (|unntities p, yj — l, |) — :2, ...(/> — / + 1). ■\rnltij)lyin<i- e<|u:i- 



tions (51) by pjf'~\ /)(/>— 1)//'"",. . . in order und tiddinu', we obtain 





For /) = -^, we hîive 



;• ! 2y— 1 ax" 2/? — 1 dx" ) 



d"ir 

 If ?/ be n rntionnl iuteofrnl function of .r of the tjn'rd deoTee, -r^ 



,-, -^ dx" 



v:iriishes for nil values of r for whicli 'ir < //. In this case, denotino- l)y 

 / eitlier — - or the intei>er next nbove -7— according- as // is a inulli])le 

 of P) or not. we have 



,r., d\/Tj_ _ j^^^j^i, .-,j!^ jr!_ iV . 



^^^ dx" ~ 2-'\n\f ?/"--• r ' /! 2/-1 dx"^'- 

 Aofain 



+ {-\r\ 1.3. ...(2r-3)(2/yr'-A;+. ..+(-!)". 1.3. ...(2/?-3)X,| , 



* Crelk-'s Journal, UJ. XXXIII. 



