MULTIPLICATTOX OF ELLIPTIC FUNCTIONS. 



155 



(3) 



-^- + n = , n odtl, 



s 



and, writing P + Qs in full, 



we must huve, denoting differentiation with respect to z by D, 



K+«i:+ «2:-+ ••+ 'A«+i c^+i + ^v^ + öi('?:) +...+Ö»-! (^r-')=o, 



«1 + «22.? + . . . + «,„^1(7». + 1 ).f" + h,Ds + h,D{sz) + . . . + 5,„_iD(s^"' -1) = 0, 



(4). 



/;„ £>-'".'? + b^D-'\sz)+... 



Since «o? ''^n ? ^i? ^ ••• ^^^ ^^^ ''^^ vanish, the determinant obtained 



by eliminatinof a^,, «,, , h,„ h, ... must vanish, that is. 



(^^) 



1 f «'2 



1 f Z^ 



5, 



f^2 



.9,r, 



1 , 2,?, (;;i + 1 )z"\ Ds, Z)(.s.?}, Z)(,s-.2-), 



.jD(.s,ï"'~») 



= 0. 





Let the expansion of this determinant according to the elements of 

 the first row be written, 



