ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 355 



where 



S = \4-^ + p— a— /3— y. 



The reader will find this important theorem, which appears to be due to 

 Richelot, fully demonstrated in Schellbaeh. If we put /Lt=/3 and p=y in this 

 equation and at the same time put y for X—a, and .i- for a — x, we have 



e.xOj/ ^-"^ sin (x + sriy ^^ 



putting in this .i- — |n for x and y—hv'i for ?/, we have 



• tixdy - -»sin(.r+(.5-i)^i) • • ' ' K.^) 

 and similarly 



0^x6,1/ -■» sm (.r + si'i) ^ "^ 



e,'o.i^(^±^^2" -^^^^^^ ... (4) 



63X di/ -" cos (a-+sj/) ' ^ '' 



6,0. ^^ ^-^ Z — V ^ . /RN 



' d,xdy — -«cos(a? + (s-i)w) • • • W 



f^iO.-^^^^ — ^■-'^=2 ^^ 1 1 .... (6) 



a^xd^y -"sin(a-'+si'i) ^ ^ 



0,xd^y *"- ■» cos (a- + si-i) ^ '' 



' 0,x6^_7j -» sin(.i' + svO ^^ 



e^6,?j - '^--^ cos(^+s^i) • • • ^^^^ 



From these important theorems the series given by Jacobi in his memoir 

 " feur la Eotation d'uu Corps " are easily deduced. If we put w=0 in 

 series (2) and reduce, we obtain 



6,0 d,ofa = 4sin x^^ q^^ia + r+') 



" 1 — 22-'s+icos2j: + ^4»+2' 



and by similar means pointed out by Schellbach ; 



60 e..offx=4 .n..v- (-l)Y+*(]-^2.+ .) 

 " 1 — 2q^^+ 1 cos 2x + 24«+2" 



eo 6,0 iu-=dy--s sin= .t-s"— (r-i)V*+' (1 +5^'+^ ) 



(l_22*^ 1 )(l-2rf^ + l~^2:v + q4s+-2y 



Oo 6,0 ^^ = cotan x^4 sin 2.rS ' (-1)V^__ 



/"'' " I — 2q^« cos 2.v + q*s' 



Oo 0.0 



lix 



.-^-J- +4sina.2" JldMI±7!!)_ 

 /r sin .6- 1 1— 22"cos2j' + 2<^" 



