MULTIPLICATION OF ELLIPTIC FUNCTIONS. ^19 



Hence, if we put 



IK = 2p 



then 



so that we need only determine B^,,, and this only for the initial 

 values of?», in view of (154). 



Now (134) may easily be modified in such a manner that the 

 resulting equation is satisfied by B. We find 



(155) |l-(H-A-V+A:V}-^+ I [{2n'-l)k'-S]x~2{n'-2)k^x^\'^ 



whence follows 



(156) (2wi+l)(2wi + 2)02^+2+1 [7r-(2m + l)2J +im{n'-vi)Ji'}B,^ 



dB, 



elk 



+ 2?^2A-(l-Ä-2)-^ + (?i2_2w)(n2-2m+l)Ä-2i5,,„_2 = 



Further we find, for m^p, 



(2w+l)(2w + 2)ß2.+2,o +{'^'-(2w+l)2]ß2„,,o - 0, 



■^2m,0 = , ?i— 1 <C 27)1 ^ 2,p, 



(2wi + l)(277i+2)J52„,+2.2 + {5n'-{2vi+\y-}B,„,^, 



+ ém{7i^-m)B,^^, +(??2-2m)(^/2-2w+l)ß2m-2,o = , 

 (2m + l)(2w + 2)52^+2, 4 + [9n2-(2w+ \)'}B,„^^, 



+ 4[(w-l)n^-m2]S2.,2 +(n^-2m)(«2_2m + l)B2,„-2,2 = 0, 

 generally. 



