MULTIPLICATION OF ELLIPTIC FUNCTIONS. 213 



whence follows 



(140) ., = (-ir "'"'-^"^-;';,/;f-'^'-^'% -(».-(2>-i)). 



For ?H = 2, we h;ive 



'^ + i^M^-ö^-)F, = {bn^-'ln'ä-\-Ö^)P,-x\n'-ö){n'-&-l)P^. 



Since the lowest power of a; in l\ is 5, we write 



(141) P, = 2«r^""^'; 



r=2 



then, 2K2>-+ l)o, + (9H2-('2r-l)Vr-i 



= ( (2r-l)2-7i2(4;--7) ) }'r-i-0^'-2>-+ 3)(7i2-2/- + 2) jvo , 



whence we find 



, _ n{nl-\){n?-^) 



».(/r — l)(n- — 9) 2 KN 



(142) ^ '• 



8, = M>^'-\X|^'--^)2(247y<«-ta25n^-l3757/^H 28625). 



Thus the general law is not ohvions as in the case of r, imd it seems 

 to be impracticable to proceed further in this way. 

 Reverting to the formulae 



since, by virtue of (109), 



