14 Art. 1.— T. TakenoucM : 



mod. p), each being repeated /times: when n=\, no repetition 

 is needed. Hence the rank is given by 



r=fd, if cl + lc en, \ 



r=f(n-[—~]\ if l<n<d+7c, V (27) 



r=\, if n=\. ' 



This result clearly includes both (6) and (21). 



§. 5. 

 Case IV (<Z=0,moa..p-l) 



Firstly we shall confine ourselves to the case ^ = -zrrr^^ 

 (mod. p). Let us consider the fd numbers, 



l + f,jr\ a=l,2,---d + k — l, a^O, (mod. p), \ 



i = l,2,-f, [ (28) 



f, satisfying (23). ) 



It can immediately be shewn that the behaviors of the numbers 

 (28), on being successively raised to the pth power, are quite 

 the same as in the preceding cases, provided a=f=h. If a=k t 

 Avhich case certainly presents itself since we suppose fc^O 

 (mod. p), then we have 



(l+f,n-y=l +pçr + - p ^~ 1) g; x*+ +ff jt*» 



=i+{p+?r 1 - Uj " 1) }? 1 - v + J3 ^~ 1) ^+ • 



Since d=fc (p — 1), 



(1 + ?^)»=] + {p* +<f+ *i }, a^feO, 



whereby the condition x^O is to be particularly noticed, and 



r "-* 1 

 consequently e ki ée h — pL <* J . Also, by the same reasoning, if all 



the exponents e' M be divisible by p, denoting by p I+!7 (<?^0) the 



lowest of all the powers of p contained in them and putting 



^'ki—p l+a Cm, we get 



