MULTIPLICATION OF ELLIPTIC FUNCTIONS. 205 



Eqiintiiig- the coefficieijts of the highest power of c (that is ç"'+''+c^ on 

 the two sides of equation (Hi)), we obtain 



ri = '2'^'"~^^"-^=2~^~, '-^ odd, 

 or, writing n instead of /^ + !2, 



n (n-3)(n+3) ., . _ . ^ \ 



Thus we find, whether is odd or even, 



Li 



(120) /; = 2^'"~^ = 2"^('^"~^^ 



and thence, when — -^— is even, 



(121) H^_i= -2''-^(;i- l)7ic^"^'^-i>-2-2^-^».(// + l)ç-^"("-i^+2 + 2^-2f^'*("^'«, 



(122) £ _i= 2^-2ç^"^"-^'--^-2^-^;H/«+l)ç*'""+i'-^-2^'-^(//-l)/?ç^"("-^2)+i^ 



and, when —5— is odd, 



(124) ^,^_i = 2^-='(;i- l)«ç5"(«-i)-3+2''-^;i0i+ 1)^-"^""^^"^-'^*"^^^^"^"+^^^^ 



Consider next -/f^_2) whereby we suppose —3— to be even. H^_2 

 satisfies diiferentiid equation (117) 



where 



ü,,_2=-2^^(/i-l)H(>r-;i-4)(/^2_vi.-6)f^"(''-l^-*-2^-^/A^M"-2X/^-y)r^^^^^ 



- 2^^''';?<?/^ + 1 )(nH /i^4)(;?2+ y;-(3)f^""^-i)+*+ 2^-*yA(;i + 3)(7t'+ 8;f^2)ç^"<"+=*)-2 



+ 2''-*/i('i-3)(M-- 3yi - 2)ç'"'"+=*^+2. 



