lt)3 R. FÜJISAWA. 



By division, we obtain 



n odd, n even, 



xA (x-) x\/i — x-'s/ 1 — A '-x'-A {X-) 

 sn nii = — ï^ — s—, Sil nu = 



(104) cil nu = 



du na = j^^^s , du na = 



D{x^) 



B{.r) 

 D{x') 



G{x') 



A.'-') ' D^x'-)' 



AVlieii n is odd, ./, />, C, i> are ail of the degree n^—\ iu x. and 

 when /< is even, A is of the degree /r — 4 wliile ./, />, C' are all of the 

 degree /r in x. Moreover, A, B, C, 1) are rational integral functions 

 of Ä;^ whose coefficients are integral numbers. The coefficients of the 

 highest power oï x in A, B, C, D are respectively 



or 



(-1)2 A; 2 , Ä; 2 , h'^ , (-1)2m A- 2 , 



(-1) ^ n k ^ , A-2 , A-- , (~1)'^ A--", 



according as //, is odd or exeii. 

 Write 



A - i'A,„,x'"^ ^ :i'/i„„(i-.-V' =- :i'.i„„(i-AVr, 



(1U5) 



^ = ^' C,„,r^"' ^- -■ C.„X1-.^'^)"' -^ ^' C,„x\-Pxr, 



then, we liave the well-known relations* 



* Compare the work of Briofc et Bouquet already referred to, or Baebr, Sur h's funiin/cs 

 pour la multiplication des fonctiona elliptiques de hi pn'ini'^Te exjièce, Uruuert's Archiv (h'r 

 Mathematik, Bd. XXXVI, pp. 125-176. 



