ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 



also 





du 



j^K; 



dlog^A.l(u,+2K,. . . .)-dlog^Al(u+-K,. ...) 



= -2,1 J. 



I , (I log, «Z(m,. . . . )a 



+ 



cZif 



lc?tt^, 



or 



d log, A1(m,+ 2K^ ....)- cZ log, Al(tv6, ....)=- S,2J^f?M^ 

 ■whence we see that 



a a 



AlK + 2i,. . . .) = (-ire~^^'''^''^"''+^''WK. . . .)• • 

 Put nj + 2Kj for w^, and remember that 



a P a /3 j3a a^ aa jS^ j3a 



and we have 



a P 



A1(«, + 2K, + 2K,....) 



= (_l)-+/5e~^^'^^^''+^^^^"''+^*'+^''^+^-^^^''~-^'''^^''^Ul(«, ....). 

 Interchanging a and /3 with one another, we have 



a p a p aj3aj3 



e — e , 



whence 



where yu is an integer. 



Section 6. — It may indeed be proved by direct integration that 



a fi a p _o 



^(V,-JKJ=±^ 



2' 



343 



(1) 



(2) 



(1) 



where the upper or lower sign is to be taken according as a is greater or less 

 than (3. See on this subject a memoir by Brioschi in the ' Annali di Mate- 

 matica,' vol. i. p. 12, in which the method of treating theorems of this nature 

 by direct integration is fully discussed. 

 The following formute are also true : — 





(2) 



