20 Art. I.— T. Takenouchi: 



Xj>0. Ttnen /t=- JlL =-r=e,i + i =« ai ; hence if we reject from R the 

 P P 



number 1 + ?X* + ' C or such a number 14 £X* for which e„i=fo. still by 

 means of the remaining fd numbers can be represented p f( ™ -1) 

 elements which, however, may not be all different. In fact, we 

 can shew as follows that some elements of 21 are not represented 

 by them. 



Now, it is evident that all the p f(n - l) elements of 31 are repre- 

 sented by the numbers of the form 1+ [p"}, a=], %, •■■ n, there 

 being <p{v n ~ a ) incongruent (mod. p") values of {p*} for each value 

 of o. It can be shewn that 



I>(p»-' , ) = l + lp«"-=- 1 )(/-l)= ?) «"- 1 ). 



l,n l,n-l 



This premised, let us first try to reject r+? < ,w ä+ * J and consider 

 how many numbers of the form l + {p' i+ *} can be represented bj r 

 means of the remaining fd numbers. A little consideration will 

 show that, to represent the numbers of this form, we have to take 

 the exponents as follows: 



e' ai =0 (moâ.p j « +1 ), if a<fc, 



e'„ i = (mod. p), if a>fr, ( 



V *' > (36) 



6'„=0 (mod. p), ( K J 



(»=1,2,...j0. ' 



Observe hereby that, for at least one value of i, e' ki ^0 (mod. J5 2 ). 

 The number of combinations of such values of exponents is 



\ pa + lj\ p J\ p p*J 



// H e ni f. 1 \ 



pHk-lHfd ' \ pf J 



= yi«-<*-*-i>(p/_l) =? ,(p»-a-*). 



But, since we are now considering the case where (31) has a 

 solution, the above combinations of exponents will give rise to the 



