Art. 5. — -T. Takenouclii : 



By the same reasoning, since 



is an algebraic unity, when /^ contains more tlian one distinct 

 prime, so must also Ije 



.(1 + %>)>/' 

 Lastly, to determine the nature of 



K(l+X>)v)' ^' ~ ^' 



when li- is a power of a single prime, observe that the three roots 

 of the equation (2) are 



which are so related tliat 



X1X.2+ x._,x^+ x^Xi = 0. 

 Therefore we get 



1 + 2.7/ \ l+2;>/ 7(lt) 



, (0 \ . [ (O \ 1 



Putting for u. in these relations a fraction in Iclf), whose denomi- 

 nator is a power of a single prime relatively prime to l + 2^o, we see 

 that 



U(0 



^(l + 2.>;«. 

 is a divisor of /-«, and its square is associated with the reciprocal of 



Thus, the number 



(l + 2;>)« 



