32 Art. 5.- T. Takenouchi : 



Corresponding to these subgroups, there are as many relatively 

 cyclic corpora of relative degree <7''"\ whose relative discriminants 

 contain no other prime than q, 



§. 10. 



The primary integer /-« has hitherto been supposed to be rela- 

 tively i^rime to 3. In the present section, let us consider the 

 case where /^ is a primary divisor of 3: 



/^ = l + 2,v. 

 If we put 



iO 



then, on account of the relations (5) and (7), it follows that the 

 corpus K{x,^ can be composed of tw^o corpora Ä"(^^(i';,)) and K{^'{v,^). 

 From the multiplication-formula 



S'{(l + 2,,)«}=M^|, (14) 



we find that 



îp(r,) = and ^-(tg'=r/3. 



Hence, if we put 



Vk 



-^Oo 



where ^^ = _^^ 



then, since the corpus K(X\Vi)), Ji>\, must necessarily contain the 

 number ^(^'s), we get 



The relation (14) gives 



y^ + ^vn-m -1 = 0. (15) 



Differentiating (14) with respect to u, we obtain 



(i+2o)îp'{(i+2o).] = ^^l^n^o^ 



which gives 



-3(l + 2,«)!>-'{(l + '2/.W=-|||^!f-'(«). , (IC) 



