The theory of determinants in the historical order of development, by Sir Thomas Muir.

DETERMINANTS IN GENERAL (CAYLEY, 1851)

67

It is thus seen that, taking for shortness' sake only three
characters, we have
(V123) = V123 + V231 + V312~- V132 - V213- V321
(V23) (V31) =-(V )=....
(V13) =0.
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 permutant whose originating symbol is V1234 is
'1234 -V2134- V1243 + V2143'
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 V1234 accompanied by a verbal explanation
as to its sets, we might write
V12
34
and so obtain
(V12 = V1-V21-V2+V 21'
\ 34/ 34 34 43 43
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
Va /3 y....
a B' y'....
One case of this is that in which it is viewed as a function of
the symbols V,.,.., VaY,... etc., and a less general case that in
which it is viewed as the,)oduct. Cayley then proceeds:"Upon this assumption it becomes important to distinguish the different
ways in which the blanks of a set are distributed in the different lines
and columns. The cases to be considered are: (A) The blanks of a