On the Relatively Abelian Corpora. ^[ 



In virtue of tlie relations : 



2 sn?^, cnitdnu 



üir2u 





l + .i-^W^i 

 1 





q(u) = 



sn^u 



it can l)e concIiKled tliat ^ii'2// can 1)0 rationally expressed by />(?/) 

 and q(ff) in the corpus kQ'). Therefore, if we put 



then the corpus JC(x,,.i) must a divisor of IvQ//,, z,), i.e. of K{_y/). 

 But, on the other hand, this latter corpus K{i/i) is evidently a 

 divisor of K{xi). Thus we find that the two kinds of corpora 



K{x^ and AXyJ, ^ = 1, '2, , 



can be arj'anged in such a series 



-^%i)' -^^X^i)' l'^^UÙ^ ^A-A'X i'^iUi^y K(x„), , 



that each corpus is a divisor of the next one. It Avill also be 

 proved presently that none of these corpora coincide witli the 

 neighboring ones, except 



K(!j,) = K{x,) = ]c{o). 



Hence, in the following, we shall consider AX///,) instead of K(x,X 

 where i/,^ is a root of the equation of the 2'''"^th degree: 



fi,-i-!Jh-i = 0. 



By (l^l), tliis equation can l)e reduced to a series of the follow- 

 ing l)iquadratic equations: 



ul-^!J,-i!jl + ^!J, + ^!Ji,-i = 0, (24) 



^ = 2,3, ,h, ,j,= l. 



The irreduciljility of (24), excepting the case k=2, follows at once 

 from that of (23). It is, however, impriniitive ; for, any one root 

 of it can be rationally expressed in /t;(/') in terms of any other root 



