ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 349 



But 2 A2,0.. A ^K;. ,«,= 2,{ 2,2^G,, /,, ^K',. Jm, 



TV 



= (by 1 and 3 of section 8) \,{J'c. a"" o^". ^^"c' 

 and therefore a,W\r^G^,^h.^^ .,k^'o='^W'\^' c i + '-/'o,2+'-3J'..3+ • • • • 



Similarly, 



Lastly, 



2^2^ Am('^^V,i + »-.K',2+'-3K;.3+ • • • •)0\K;i+ . . . .) 



A V r, c, ij. 



A A' ■^ 



= 2{ VaJ'o, a VvK'., v} - i WA'VSeGe. A^'., V 



Whence we have 



From this expression, combined with that given in last section, we may 

 develop Jc{v^v,^v^ . . . . ) in a series of exponentials. The full expression is 

 given by Konigsberger, Crelle, Ixiv. p. 19. 



Section 11. — Hitherto our investigations have had reference chiefly to 

 whole periods. We wUl now investigate some formulae involving half 

 periods. 



a 



To determme Al(tf^ - K^ . . . . ). 



By a former equation, we have 



a ^ ( i d log al(u,u., ....)„] , 



cHog, AiK+K,. .)-cnog,Ai(iv.)= -2.| J.- °^ )j;^^' — -^yK 



