On the Relatively Abelian. Corpora. 7 



If « be an integer in h{(') and relativel}' prime to 3, and if /^ l)e 

 I'olatively prime to «, then the nnmbers 



\ a 



are all associated with one another. 



If /^ be a power of a prime, then the reciprocals of the 

 numbers 



_/ vco 



\ fi 

 are divisors of [J-. 



If /^ consists of more than one distinct prime factor, then the 

 numbers 



^\ // 



are algebraic unities. 



Next let us consider the case where i*- is not relatively prune 

 to .']. In this case, y- is necessarily divisible by l + 2o, for 



3==-(l + 2.v- 

 Since 



-Kl+2/'(a") — -P'— 1' 

 -t i_j_2^) \X) —— OX ; 



we get 



x'' + 37jx- — l = 0, (2) 



where x = t(ï{), y = t[{\ +2f>)u]. 



This equation shews that x is an algebraic unity, provided that ?/ is 

 an algebraic integer. ' 



Now it can easilv be seen that 



1+2/^ 

 Hence, by the repeated application of (2), we conclude that 



Vco 



(i+2.y 



are all algebraic unities. 



], A->I, 



