75 



I cannot find that either Cauchy or any of his followers 

 have ever suspected that on the group 



12 3 4 5 6 



2 14 3 6 5 

 can be constructed any grouped group of six except this, 



12 3 4 5 6 



2 14 3 6 5 



3 4 5 6 12 



4 3 6 5 2 1 



5 6 12 3 4 



6 5 2 14 3 



It is laid down in one of the last French contributions to 

 this theory (vide pages 22, 32 of " Theses presentees a la 

 faculte des Sciences a Paris, &c.," Paris, Mallet Bachelier, 

 1860), as a point apparently too plain to require demon- 

 stration, that there are in a grouped group two species of 

 substitutions, ^^ deux especes de substitutions, hien distinct esi"* 



1. those whereby the elementary groups simply change 

 places among each other ; 



2. substitutions whereby, the elementary groups being 

 undisturbed, displacements of letters are effected exclusively 

 iij their interior. 



The truth is, that there is a third species of substitutions 

 which occurs in grouped groups more frequently than the 

 other two, whereby both the elementary groups are permuted 

 and also displacements of letters take place in their interior. 



For example, there exist the three following groups con- 

 structed like the preceding on the partition 

 N=6=2-3=A«. 

 123456 123456 123456 

 214365 214 365 214365 

 346512 435612 345621 

 435621 346 5 21 436512 

 561243 651243 562143 

 652134 562134 651234 

 in which there are substitutions of the third species* 



