448 
Proceedings of Royal Society of Edinburgh. [sess. 
“De plus, ces memes suites ou arrangements se partageront 
en deux classes bien distinctes, la comparaison de chaque 
nouvel arrangement au premier 
a, b, c, d, ... . 
pouvant donner naissance a un nombre pair ou a un nombre 
impair de groupes.” 
Of course, tbe primitive permutation is looked upon as having its 
groups also, viz., one for every letter in the permutation. 
Then comes tbe important proposition — The interchange of two 
letters increases or diminishes the number of groups ( substitution- 
cycles ) by unity. In proving it the two letters are first taken in 
different groups, 
( a,b,c , . . . , h,k) f ( l,m,n , . . . , r,s); 
and since any member of a group may occupy the first place, the 
letters a and l are fixed upon. Now what the groups imply is 
that the letters 
a, b, c, . . . . h, k, l, m, n, . . . . r, s 
in the primitive permutation are changed into 
b, c, . ... k, a, , m, n, . ... s, l 
respectively to form the given permutation. If therefore in the given 
permutation the letters a and l be interchanged, the new permuta- 
tion so obtained will be got from the primitive by changing 
a, b, c, . . . , h, Jc, Z, m, % ... , r, s 
into 
b, c, h, l, m, n, , s, a ; 
that is to say, by the changes indicated by the single group 
(a,b,c, . . . , h,Jc,l,m,n, . . . , r,s). 
The interchange of two letters belonging to different groups is 
thus seen to reduce the number of groups by one. On the other 
hand, it is clear that had this single group belonged to the given 
permutation, the interchange of two letters, a and l say, would have 
the effect of breaking up the group into two, 
( a,b,c , . . . , hf) and (l,m,n, . . . , r,s). 
The theorem is thus established. 
(hi. 35) 
