81 
(35) Theo. E, If N — 1 be any prime number, we can 
form t r (N — 3) equivalent groups, each of N • (N — 1) (N — 2) 
substitutions. 
The enumeration of these groups is, as I believe, new. 
(46) Theo. F. If N > 5, it is impossible to construct a 
group of N • (N — 1) (N — 2) (N — 3) substitutions, which 
contain N powers of one substitution. 
(53) Groups of the form G-\-RG, R G being composed 
of square roots of unity , 
Theo. G. Let 
N=Aa+B6+Ce+ • • • + J/ 
when A72 ; A 7 B 7 • • 7J, and where one at least of ah’ 'j 
is 7 I. 
Let K be the least common multiple of A B C • * J, and 
let Il A - be the number of integers (unity included) less than K, 
and prime to it. 
There are 
7tN 
R a K 7ra ' Ttb ' ' ’ rcj 
equivalent groups of 2 K substitutions of the form Gq-R G, 
where G is the group of K powers of a substitution having a 
factors of the order A, b of the order B, etc. ; and where RG 
is composed of K square roots of unity. I believe that this 
is new. 
For example, take the partition 
G=3-2=A«. 
There are 20 groups G by theor. C ; and 60 groups G-f RG 
by theor. G. Let G be 
1 2 3 4 5 6 
4 6 1 3 2 5 
3 5 4 1 6 2 
B 
