81 



(35) Theo. E. If N— 1 be any prime number, we can 

 form TT (N — 3) equivalent groups, each of N • (N — I) (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=:A«+B5-l-Cc-i- • • • + J/ 

 when A72 ; A7B7 • • 7 J, and where one at least of a 6 • *j 

 is 71. 



Let K be the least common multiple of A B C * ' J, and 

 let Hk be the number of integers (unity included) less than K, 

 and prime to it. 



There are 



ttN 



equivalent groups of 2 K substitutions of the form G-}-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 R G 

 is composed of K square roots of unity. I believe that this 

 is new. 



For example, take the partition 



6=3'2=A(/. 



There are 20 groups G by theor. C; and 60 groups G+RG 

 by thcor. G. Let G be 



12 3 4 5 6 - 



4 6 13 2 5 



3 5 4 16 2 

 1? 



