1904 - 5 .] 
Dr Muir on General Determinants. 
937 
In his paper of the year 1847 an altogether different generalisa- 
tion 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 ‘ permutants. 1 For the sake of easy 
exposition we shall follow him in his special usage of the words 
‘ form, 5 * blank, 5 4 characters, 5 1 symbol. 5 “ 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 5 
(previously called by him * nombres symboliques ’) be 1, 2, 3, 
4, ... . the ‘ symbol 5 may always be represented in the first 
instance, and without reference to the nature or constitution of 
the ‘ form, 5 by V 1234 . . . ; for example V 1234 . . . may stand for 
-Pl2Q3-^4 • • • J 0r ^12-^34 • * • j 0r • • ' • 
How, 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 permutantd 
The originating symbol being V 1234 , the corresponding 
permutant might have been denoted by % ± V 1234 ... as in his 
paper of the year 1847, but as a matter of fact Cayley now makes 
use of 
(Vim. ..:)•• 
It is thus seen that, taking for shortness 5 sake only three 
characters, we have 
0^123) = ^123 ^231 + V 3 12 — ^132 ~ ^213 — ^321 > 
(V 123 ) = (Y 231 )=-(V 132 )=-**- 
0^113) == 0 • 
