ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 319 



Section 8. — Kosenhain points out that by means of the Table he is able 

 to obtain thirteen out of the fifteen ratios ^-^ in terms of any two of them. 



He selects for that punjose the ratios ^ '' . and ^^^' " , ' — ; ; he then 



introduces the new variables a\ and .r^, and assumes 



T'i ? \ ^ — "-AU . X,X,, 



where 



Z,-^^ ^ 2. 2 ^ 2. 3 X2^ ^2, 0'2. a 2 0"2. ^''2, 3 



^"".1 x2 ^'n, 'P'n, 1 2 0^. 0°o. 3 . 



z,- 



0\. 



^ 3,2^3,3' ' 0^00%, 2' ^' 1>\o1>\.' 



whence it follows from equations (C) that 



and from equations (A), that 



0% , (v, w) \u 



0o,o(^'^) A'iMW 



X» 1.2 \2 2 7.2 2 2 \2 2 



^=k-\, fx^=lc-ix, fx.=\-n. 



where 



Rosenhain also (p. 423) shows how the remaining ratios are to be found. 

 I shall write down three of them, denoting 



.r(l -x){l - F.t;)(l - \\v)(l - ix\v) by E(.r). 



+ 



1 (l-XX)(l-/ag - (l-X^r,)(l-/t^r,) 

 0!i,^i!i!i) ^ A(l-.rJ(l-.r, )( l-\X) ( l-A'-^-.) 



(] -,rj(l-\^r,) - (l-.r,)(l - AX) J ' 



