MULTIPLTCATTOX OF ELLIPTIC FUNCTIONS. ^77 



^2 ^2 ! f?.r" ^ d.r" ' 



(51) 



'" ~ v! c/^r" (/•-!}! 1! cZj-" "'*(/•- 2)! 2! fZ.r" •• 



+(-ir^ 



l!(r-l)! f/.r" ' 



»~ 71 1 d:r'' («-1)!1! ^;r""i (n-2)!2! dx" 



\ ^^ ^ l!(7i-l)!fZ:r»' 



It may be worth while to notice that the expression on the opposite 



side of A', is identically zero for ;dl values of r oreater than n. 



From the mode of derivation it is evident that A,, contains 



1 d'^dr) 

 differential coefficients of?/ but not ?/ itself. Hence, if — j — '^^ be 



./ ./ ' r ! dx" 



broken u]) into two parts such that one part contains all the terms 

 independent of explicit // and the other part, terms having y as a factor, 

 then 



1 d"(u') 

 X,. = that part of -^j j^ which is independent of explicit y 



1 d'KiD 

 and that ])art of -^ "77 n which contains explicit // together with 



y cny'-') y' d^y"^^) , , -.y, f ^^y 



(r-l)!l! dx'' "^ (r-2)!2! dx"" "^ ^ ' \\{r-\)\ dx" 

 is identically zero. Now 



V ! dx" 



rocffioieiit of /(" in \i/-\-Ji^ + -rr ^,+ ■ ( 

 (■ dx 2 ! dx- ) 



