54 Art. 5.— T. Takenouchi : 



where Bi< and (7s are integers in k(p), and divi8i)>le Ijy r. Intro- 

 ducing the relation 



r, r- =r> 



and also remembering tliat 



(p-l)"-' = l (mod.-). 



we find, after a few steps of transformation, that 



(,-n^(.„) = - ^^^;^f+'? . (20) 



where (^ and it are rational integral functions of ^-(i(), the coefficients 

 being all integers in k(f>) and divisilde 1>y -. 

 On the other hand, if we put 



■we get, from §, 1 and §. 2, the formula of the form 



the coefficients a' s and the a' s being integers in lip). 



We have alread}^ shewn (§. 3) that the <^'s are all divisible 

 by -. XoAV, comparing (27) with (26), we see that the «'s must 

 also be divisible by r. It follows therefore that 



zijziif = zi'ufi' (mod. -). (28) 



Hitherto we have supposed that ô is even. But, if it be odd, we 



may use [>- or [>-- instead of - ; for 



f>- = -h-{-{a-b)o, 



f>-- = —a + b— ap, 



and either a— h or a must certainly be even. Therefore the relation 

 (28) holds good, even if b be odd. 



This premised, we are going to shew that, if p be an integer 

 in Ä(,o), and 



O) 



U = 



