75 
I cannot find that either Cauchy or any of his followers 
have ever suspected that on the group 
1 2 3 4 5 6 
2 1 4 3 6 5 
can be constructed any grouped group of six except this, 
1 2 3 4 5 6 
2 1 4 3 6 5 
3 4 5 6 1 2 
4 3 6 5 2 1 
5 6 1 2 3 4 
6 5 2 1 4 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, Men distinctes;' 
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 
in 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«. 
1 2 3 4 5 6 
2 1 4 3 6 5 
3 4 6 5 1 2 
4 3 5 6 2 1 
5 6 1 2 4 3 
6 5 2 1 3 4 
1 2 3 4 5 6 
2 1 4 3 6 5 
4 3 5 6 1 2 
3 4 6 5 2 1 
6 5 1 2 4 3 
5 6 2 1 3 4 
1 2 3 4 5 6 
2 1 4 3 6 5 
3 4 5 6 2 1 
4 3 6 5 1 2 
5 6 2 1 4 3 
6 5 1 2 3 4 
in which there are substitutions of the third species. 
