On tlie Relatively Abelian Corpora. 13 



consequently 



o snhi S,? 



Hence, if we regard /S',„ as a function of .T = sn?^ and express it by 

 S'(i-l-i-,;). ^veget 



-l^.,A,.{x) = sJX- /^ ). (9) 



In the limit when a-=oc>, we get 



lv.- = .sv(-^,!,). 



The value of the right-liand side can lie calculated successively by 

 means of the recursion-formulae of /iV given in §, 2. (The supposi- 

 tion that h is even is essential in this calculation.) Tlie general 

 form of it is 



1 V nArl-\ h 



Therefore 



£ = C-iyy-V^'""' = ^"-' 



and the product of all tlie roots of the equation 

 is equal to 



(-i)V'-v. 



Now, it is Avell-known that, if ^wu ha any one of the roots of 

 this equation, tlien l)Oth cn?^ and dn?/ can l)e rationally expressed 

 by sn?^ in the corpus /r(/'). Therefore, it follows from (G) that the 

 corpns ol;)tained b}' adjoini]:ig sn?^ to Z-(/') must remain unaltered, if 

 we change p- into another integer associated with i'-. ( )n the other 

 hand, it is certain that, if there be given any integer with the odd 

 norm in ^{z'), then among the integers associated with it Ave can 

 always find one, and only one, possessing the form (8). Hence, 

 in treating the nature of the corpus /•(/', sn?/), wo may entirely 

 confine ourselves to the case Avhere /'• is of the form (8), provided 

 that the norm m lie odd. 



