80 
I believe that this enumeration, though I think it a little 
surprising, is new. 
(11) Theorem C. Let N=Aa+Bi+Cc+ . . + J /> 
A 7 B, B7C, . . F7J, a, b, c, . . .j being any integers 
- 7 0, 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 . . . . j circular factors of the order J, is 
w- v N 
lt A - Tra'Trb * * • irj ’ A a ‘ B l * * • J ' ’ 
where R* is the number of integers less than K and prime to 
it, including unity. 
I believe that this enumeration also is a fait nouveau : and 
it is fundamental. 
(16) Theorem D. If there be, among the R v — 1 integers 
less than N and prime to it, a prime root of the congruence 
x r — 1 = 0 (mod. N), 
where r ^ R v , 
we can form with N elements — — equivalent groups 
\x N 
(l 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' ^ * ^ different equivalent groups 
R.v 
of the same description. 
(18) Cor. With the N elements we can form ^ - — D 
ily 
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. 
