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

66 HISTORY OF THE THEORY OF DETERMINANTS

the one important point to be noted is the relation between
hyperdeterminants and the functions of Cayley's paper of the
year 1843, and this is shortly indicated by saying that the
latter functions are hyperdeterminants in which p=1 and n
is even.
In his paper of the year 1847 an altogether different generalisation was formulated, the corresponding symbol being
+_(1, 2,.,., ),
and one of the objects aimed at being to extend the definition of
a determinant so as to include within it the Pfaffian. (See
Collected Math. Papers, i. p. 589.)
Having thus attempted to make clear the stage which the
process of generalisation had reached with Cayley prior to 1851,
we are prepared to appreciate the notable advance made by him
in his paper of that year. The widely embracing conception
therein formulated was that of the functions called on the
suggestion of Sylvester 'pevrmutants.' For the sake of easy
exposition we shall follow him in his special usage of the words
'form,' 'blank,'' characters,' 'symbol.' " A form," he says, "may
be considered as composed of blanks which are to be filled up by
inserting in them specialising characters, and a form the blanks
of which are so filled up becomes a symbol." If the 'characters'
(previously called by him 'nombres symboliques') be 1, 2, 3,
4,.... the 'symbol' may always be represented in the first
instance, and without reference to the nature or constitution of
the 'form,' by V1234...; for example V1234... may stand for
PQ3R4..., or P1234..., or....
Now, let the characters 1, 2, 3, 4,... in such a symbol be
permuted in every possible way, and the resulting symbols have
the sign + or - prefixed to them in accordance with Cramer's
rule, then the aggregate of all these symbols is a 'simple
permutatt.' The originating symbol being V1234..., the corresponding permutant might have been denoted by Z ~+V 234..
as in his paper of the year 1847, but as a matter of fact
Cayley now makes use of
(V1234....).