MULTIPLICATION OF ELLIPTIC FUNCTIONS. 221 



§ 1«. 



Consider next the ca«e where n i.s even. To begin with, take 

 sn nil. Here D is exactly of the same form as in the case where )i is 

 odd, so that we may restrict ourselves to the consideration of A alone. 



Now A is of the degree y^^ — 4 = 4^ say, and 



(159) A {X, k) - (- 1)'^' A {-^, k\ k-^ x'", 

 and, therefore, 



(160) A^^,„, = (- 1)"^ A,,, k'^"\ 



m - 22) 



In consequence of (104) and (134), A = ^ A,„^x^"' satisfies the 



m = U 



differential equation 



rPA flA 



(161) |^-(l + Ä;V+^'-^'}-Tl+ 1'-^+ [(2n2-5)^•2-5]^2-•2(«2_4)^V^-p 



dx^ " dx 



dA 

 dk 



dA 

 + 2n^k{l - k'')x-^ + («2-4) {(1 + k^)x + {)v'-S)k''x^]A = , 



whence follows, 



(1()2) ('2/;t + '2)('27yi + 8)^i2,„+2+ {n^-4:{ni+l)-+ [(4;/i+ l)n--4.{iii+ \f\ k-}A.„, 

 + 2n^k{l-k^)^^ + {n'-2m-'2){n'-'2m-l)k^A^,,_^ = . 



By virtue of (1()0), we need only determine the first half of the 

 coefficients A^^. Again, A.^m is of the form 



A,„. = A,,,a^ + J^n + A,,,,,A^'+^r''^-')+-+A,,,,,,A^>^'+^^^^^^^^^^^ , 



the last term being ^s™,»«^'" or ^2,», «i-i (^"''^ + ^"'^^), according as ;;« is 

 even or odd. Substituting this in (162), we get 



