938 Proceedings of Royal Society of Edinburgh. [sess. 
These preliminaries having been grasped, “it is easy,” in Cayley’s 
own words, “ to pass to the general definition of a permutant. We 
have only to consider the blanks as forming, not as heretofore a 
single set, but any number of distinct sets, and to consider the 
characters in each set of blanks as permutable inter se, and not 
otherwise, giving to the symbol the sign compounded of the 
signs corresponding to the arrangements of the characters in the 
different sets of blanks.” Thus, if the first and second blanks 
form a set, and the third and fourth blanks form a set, the per- 
mutant whose originating symbol is V 1234 is 
The idea is hereupon suggested of arranging the blanks of a 
compound permutant so as to show in what manner they are 
grouped into sets. For example, instead of doing as we have 
just done, viz., using V 1234 accompanied by a verbal explanation 
as to its sets, we might write 
From this it is a simple step to the idea of grouping the blanks in 
lines and columns, that is to say, to such a symbol as 
One case of this is that in which it is viewed as a function of the 
symbols V a py . . . , V a '/3y • • • , etc., and a less general case that in 
which it is viewed as the product. Cayley then proceeds : — 
“ Upon this assumption it becomes important to distinguish 
in the different lines and columns. The cases to be con- 
sidered are : (A) The blanks of a single set or of single sets 
are situated in more than one column, (B) The blanks of each 
single set are situated in the same column, (C) The blanks of 
each single set form a separate column. The case B (which 
34 
and so obtain 
the different ways in which the blanks of a set are distributed 
