80 



I believe that this enumeration, though I think it a little 

 surprising, is new. 



(11) Theorem C. Let N=A«+B6+Cc-}- . . + 3j, 

 A7B, B7C, . . FZJ, a, b, c, . . ,j being any integers 

 yO, and let K be the least common multiple of ABC ... J. 



The number of different equivalent groups, each being K 

 powers of a substitution having a circular factors of the order 

 A (i,e, of A elemnts), b circular factors of the order 

 B .... 7 circular factors of the order J, is 



^___ZN ^__ 



^^— R^ Ta-TTb • • • Trj'A^'BT' • • J^ ' 



where Rk is the number of integers less than K and prime to 

 it, including unity. 



I believe that this enumeration also is a /ait nouveau : and 

 it is fundamental. 



(16) Theorem D. If there be, among the R^_ 1 integers 

 less than N and prime to it, a prime root of the congruence 

 a;'— 1=0 (mod. N), 



where ;• ^ R,y, 



we can form with N elements -^-^ equivalent groups 



(I Ai Ao . . . ) each of Nr substitutions, among which will be 

 found the N powers of a substitution of the order N. 



And if there are m prime roots of this congruence of which 

 none is comprised among the powers of another, according to 



this modulus, we can form m* -X^ ^different equivalentgroups 



of the same description. 



(18) Cor. With the N elements we can form ^ jf^ 



equivalent groups of 2 N substitutions, each comprising N 

 powers of one substitution, and N square roots of unity. 



I believe that this theorem D, and the corollary, present 

 what is new. 



