26 Art. 5.-T. Takenouclii: 



^ is a cyclic proui^. If 



P--^ =^Hi^j^. ^ 



Avliero 2'^^^ P-^ ^^'^ distinct primes, the invariants of 23 are 



7>"S 7?."-, 



3( is an Abelian group, wliose invariants consist of powers of 

 21. If ^'' be the highest power of p contained in 77, then tlie I'ank r 

 of %i is 



r = fd + l or fd, if n^d + l; /.• = T-^], 



according as the congruence 

 2) + oc'~^= (mod. p''+i) 

 lias a solution or not, 

 r =fd, if n — d + J,-, 



r-/(7^-[|-]), if d + l->n>l, 



r = 1, if ;i = 1 ; 



where the sj-mbol [.c] denotes the natural number, such that 



rr + 1 r> l-c] ^ .r. 



In the case where d is not divisible by 75 — 1, we can find not 

 only the rank, but also a system of bases as follows. 



Let 7: be such an integer in the given corpus, that is divisible 



by p, l)Ut not by P", and ç;(/ = l,2, ,/) such integers in the 



same corpus that 



c^ç^ + c..ç.2+ +Cj Çy ^ (mod. p), 



for all rational integral values of c, (/ = 1,2, ,/), except when 



Cj = r._. = = cv = (med. 2^). 



Then, the following yH numbers represent a system of l»ases: 



«, = 1,2,3, • , f7 + /<• — 1 , excluding" multiples of », 



^ -1,2,3, ,/. 



Even if d be divisible by ^7 — 1, we obtain similar results, pro- 

 vided that n ^ d + k. But, if 77^>d-\-kj we cannot determine a 



