Art. 5.— T. Takenouchi : 



with respect to k{oj). But the relatively Ahelian corpora witli 

 respect to 1c{(o) are not complete^ exhausted bj^ s^ra/J-corpora. 



The chief object of the second part of the present paper is to 

 make these points clear in tlie special case in which the fundament- 

 al quadratic corpus is /"(/>). 



PART I. 



§. 1- 



Consider tlie function ^(li), Avhose periods lo, to' are in the ratio 



a) 



p, p = 



l + V-3 



For sucli ^-function, we have 



rj, = 0, 

 and consequently 



Tills îf-function admits of complex multiplication. Namel}^, 

 if we denote by /^ anj^ integer in the quadratic corpus /•(/'). then 



Hf>')=p- (1) 



where P and R are rational integral functions of ^^(«) of degrees 

 //I— I and m respectively, m being the norm of /^. Let ns suppose, 

 once for all, that tlie coefficient of the highest power of ^-(?/) in F is 

 equal to /^'. Then, putting u=0 in (1), we find that the coefficient 

 of the highest term in R must be equal to unity. 

 Let us now introduce a function c^v, !^nch that 



(/',, = n, when vi = or 1, 



where the product is to be taken for all such incongruent residues 

 > with respect to the modulus /^-, that are not divisible by «. Then 

 we get 



P = 6"- 



