THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEItf. 259 





 (6), 



m 1 



1 



l\/T)o,, 

 N " 



v-L 2 



a __~- 



<* // \-l->2;| 



is such that 



(10), 



so that JACOBI'S division-values of the Second Stage can he calculated. 



The Weierstrassian Sigma Functions of the First Stage in 4 can now he converted 

 into Jacobian Theta Functions of the Second Stage by the relations on p. 52 of 

 ' Formeln und Lehrsatze ' ; and forming the series of functions 



( 12 ). 



Til 



(13), 



a m 



-ii) A/- 7, (14), 



7 V a 7/t 



_.,-;/', ma N,, 



C., r = 05 *X 2ll+1 - -r>"" 



a m D r 



c _ jp-Jx-^T A / a N 2'-+i y (16), 



2f+1 " " V a m 1 D ir+1 72 



we find that 



rfK' Hr/K' / 17 x 



c 80 r HE' (17) ' 



or one of its linear transformations ; and <\J - is always rational in a and m, m 



consequence of the relation 



s ( w ) _ ., ( WV ) =; S (fl)) - S ('/I + 1) V (18). 



2 L 2 



