MULTIPLICATION OF ELLIPTIC FUNCTIONS. -JOT 



When — ;t— is t'Aen, 



Ug_^ = '2"~ \ii — \)n()i - 'S){n + '2)( /r - //, - «)( ir -n-l 0)ç'"^" "^^'^ 



+ '2"-'\ii, — l)n{n - S^Sn + '2){n' + '2;r'- 21 /r - 42;; + G( )lç'"("-i>-2 

 + 2f-»»(;?^ + \)(n + 8X3» - 2)(;/,*- 2;/.- - 2 1 ;/- + 42yt + GO)ç^'''"-i'+- 

 + 2P"'-'//.(;? + 1 )[it + H){)i ~ 'Dili- + II - H]{ii- + It - l0)ç^"("-i>+6 



- 2''-'n{n + 3)iii + 4:)(:ii - I )(;;2 + 8;i - G)|^"( "+«>-* 

 -2'^^'';^^(;;''-l9;r + 98)ç^"^"+='^ 



-2-''-''yt(;j- oK/? -4K;? + 1 ){ir- -yii - 0)ç'"(»-3)+4. 



A particular integral may easily be foiiiicl. I'lie cum[)leiiieiitary 

 f unction is /gÇ-"'""^', where f^ is a function of//. Hence we get, — ^j— 

 being even, 



am H,_,= f;ç^"<"-« 



_ 2P-i^^-s-in{ii - 1 )(if" - u - 8)(;r - n - lUlç '"'"-^'^ ^ 

 -2^i'V<M-8)(;?'' + 2;i^-21;r-42y/ + (U))ç^"f"-i)-- 



- 2»'-i"yM;A + 8)(»*- 2;/.^- 2br + 42;f + G0)ç''"<"^^*+2 

 -2''-^"3-^;K« + !)(''-+ '^-8K'^' + 'i-^0)ç^"^"-i>+« 

 + 2^-«;?(yi + 3)(«2 + 3;^ - 6)ç^"("+^^-* 



+ 2»-\n'- 19»H 98)?^"'"+^^ 



+ 2^-«w(w- 3)(7?-- 8?^ - 6)ç-^"("+^^+\ 



(180) ^^-^3= 2"-'VVi-3Hy^'-87/.-6)l^^"<"-^-''-' 

 + 2''-\y/*- l<.)y?- + 98)p("-''-^ 

 + 2'^-"y?V/ + 8)(/r + 3n - 6)|^*"('^-3)+3 

 _ 2P-^'K-^-ht{it + \)(n-+ii-8)(ii- + ii- 10)ç-'«<"+i)-7 



- 2P-'^''n{ii + 'à)[ii^-2n'''-2\ii- + 4-211 + OU)ç'"^"+i'-^ 



- 2^--i";/(yt- 8)(M* + 2yr^- 2br- 42/i + 60)ç''""'+i^-^ 

 -2"-^"8-^;K»- l){ii--n-3)[ir-ii— lO)ç^"('^-'^>+'o 



When —:t— is odd, we get, l)_v writing — // instead of// in (1-Î'), 



