MULTIPLICATION OF ELLIPTIC FUNCTIONS. 



175 



d^u _ du d'^ij d''n ( ^ dij d'^ij . d^ji dhj 

 dx-' dy dx^ dif ( dx dx^ dx^ dx^ 



d!^ii/dij\'dhj d'u/d//\ 



\ dx ) dx^ dy^ \ dx ) ' 



dy* \ dx / dx^ dy^ 



and, generally, 



d"iù du d'il .^^'■'S'a- I d'"'>'' Y ,'^"'W 



^^^^ -d^ ^ 7hj^'-^df^'^---^dV^'-^---^l^^'^-''^dr^'^' 



where Aj, Ag, V,^ denote a« yet unknown l'uiK^tions which contain 



only the differential coefficients of // with respect to x. A few of the 

 initial and end coefficients may at (jnce be written down : 



Y — 



d^ 

 dx' 



du d"-h/ nhi—l) dh/ d" hi , 



LttAj CLJu 



2 ! dx' dx"~ 

 ln{n-l)...{n ^ + 1) ^^ 2 ^j 2 



+ 



('¥) 



dx '' dx '' 



n(>i— l)...Oi— -IJ-+1) If-in 



(50) / 



'^ • (^) 



, dx '-^ 



, (/i even) 



A.. 



;iO?. — 1 )(yt - 2) / r?// \ "--^ rZ'V/ 



/ dy Y~'' d !/ 

 \dx J Tbâ 



+ 



n{ii— 1 )(y^ — -IXn—'^) ( dy 



(djiX'Vdj/\ 

 \dx ) \dx-) 



_ n{n.~\) /dyy-'-d'y 

 "—1 o I I ,7 ., I .7 ..2 



^A. 



/djiY~'d^ 



dî^- 



