22 Art. I.— T. TakenoucH: 



This shews that here again the remaining fd numbers are not 

 sufficient for representing all the elements of 21. 



Thus we see that none of the fd + 1 numbers in R can be 

 rejected, and consequently we have always r=fd+l. 



The results obtained in this and the last paragraphs can be 

 summed up as follows : 



If we suppose d=0 (mod. p— 1), butd^O (mod. p(p— 1)), then 

 the rank of 31 is given by 



r=fd + l or fd, if n>d + Jc, 



according as the congruence p + 7i' t x v ~ i = Q 

 (mod. p* +1 ) has or has not a solution, 



r=fd, iî n=d + k, l '"' 



r=/ ( K "[ir]) tf i<n<a+Ä, 



r=\, if n = \. 



§• '. 

 Case IV {Continued) 



In the preceding two paragraphs we have for simplicity c n- 

 fined ourselves to the case k^O (mod. p). The same reasoning 

 also applies to the case k = (mod. p), with but slight modification. 



Firstly, when n£.d + k, the invariants and the rank can be 

 determined as before without any difficulty, the result being the 

 same as given in (40). 



Next, when n>d + k, suppose that k=ap^; then if at least 

 one of e' ai (i=l, 2, ■•■,/) is not divisible by pi* + 1 , we have 



y/(l+^) e '«<=l + {p<n jéj a , 



but if e\i are all divisible by p ja + 1 , put e' ai =p?* +1+a . c t) where p ja+1+3 

 (</^0) is the lowest of all the powers cf p contained in e' ai 

 (V=l,2, ■• ,/); then 



