24 Art. I. -T. TakenoHchi : 



fl h e ■ rj-rtn-U-i 



e d+k -— -; — numbers are represented, which aie all in- 



e al pu l 



congruent to one another (mod.p"). Hence, if we consider the 



product 



ii=a+ç 1 7iif' aï q, 



it will be found as before that all the elements of 2Ï are represent- 

 ed by it, each repeated -j-xt times. Therefore we conclude 



rÉfd+1. 



That r=fd+l exactly, in other words, that no one of thefd+l 

 numbers in R can be rejected, can be shewn as in the last para- 

 graph. Only we have to make the following changes: 



In (36), replace e' 4i = (mod. p) by e ai =Q (mod. p k + i ), 

 and e' M *0 (mo,l. p 3 ) by e' ai +0 (mod. p i,t + 2 ). 



Consequently (37) becomes 



Vexcept n 1«+ l /\ P A ,»'« +1 n ^ + V 



filled ( x 1 \_ 



a<k y if )~ 



V 

 In (38), replace k£a<ß by k<:a<:ß, 



and consequent changes should be made in the first 

 two lines of (39). 

 Thus again we arrive at the conclusion (40). 



Now, the congruence 



P + jtV-'hO (mod. r.^ 1 ), (42) 



which is used in (40), contains t, while the rank r must be in- 

 dependent of the choice of 7t . To make clear this independency, 

 we may use the congruence 



p + o'-'sO, (mod. r,^ 1 ), (43) 



