ON KLLIPTIC AND HYPERELLIPTIC FUNCTIONS. 



341 



V 





A m =:: — ^^^ f\e I I 1 I t \ Xiai-'B' 



v/^ 



2v 2 2 



where 



and t\^ , &c. have the values we have just given, 



^l 



^(^>^.'^i.i»^i,2' '•a.a)! 



^,1 ^K' -^2. '■l,!''-!..' '■2.2)5 



•R'(l) 6K' "^2' '•1.1. '■1.2. ^2.2), 



^/^ 



= «?(««; + /jMj+e, yu^ + Su^+^,C,l,m)^, 



where e and ^ are two constants introduced by the integration. 

 Also 



yu+Bu^+^= 2C,^ ^v^ + 2C,^ ^v.^, 



where the quantities C are the same definite integrals as the quantities K, 

 if c, I, m are substituted for k, X, fx, and - has the same relation to C that r' 

 has to K. 



After giving a variety of formulee about the periods of the hypereUiptic 

 functions, in conformity with the notation ado])ted by Dr. Weierstrass, 

 Konigsberger states the problem of transformation thus : — 



If 



a«, +/?«,+ 2a>HKj^^ + 2^3mK,_^ + e = 2Cj,y, + 2Cj,y,, 



yu, + lu^ + 2y«iK,^ , + 2lmK.^^ , + ^ = 2C,, ^w\ + 2C,, ,w',, 

 aud 



correspondiug to the periodic system 



a l{u^ + 2Kj_ , , n„ + 2Kj^ ^ )\ = aV(u^u^)^, 



