ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 335 



J^ 



since 'Rx='R^x^ + x'Bi^x^, 

 We call also reduce 



by putting cx'^ = az''. 

 So also 



{^r=J^= (4) 



r dx R(^=) 



by putting a;^ = -, to -vrhicli can be reduced 



f. 



since ll(.i;) = Rj«--|- a.'ll..T^. 

 AVe arc also able to express 



by elliptic functions, if we put xz= ^ax-\-hx-\-cx^, to which can be reduced 

 Cdxn^^_ (8) 



J v^rt + /^^ + ex'- + f a.''' 

 e a + hp 



by putting x=ij-^p, where « + /^j^ + fjj^ + (2j'=0. 

 Again, Ave can reduce 



s 



to elliptic functions, if we make x^z^ =a -{-hx- -\- cx^ ; and also 



''"f <f (10) 



i 



if we make - = \/a + 6.r + cx'. 

 Moreover 



J'C/«+''-^''+('A'* 



can be reduced to the two last cases, since 



Lastly, the more general integral 



C dx . R{x) ,^2") 



can bo reduced to (11) by putting x=- — —, and proceeding as in elliptic 



y-\-z 



functions. 



