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

SKEW DETERMINANTS (CAYLEY, 1847)

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n - 3 derniers nombres de chaque permutation, et ainsi de suite; le
signe etant toujours +. II pourra etre demontre, comme pour les
determinants, que ces fonctions changent de signe en permutant deux
quelconques des nombres symboliques, et qu'elles s'evanouissent si
deux de ces nombres deviennent identiques. De plus, en exprimant
par [12.. n] la fonction dont il s'agit, la regle qui vient d'etre enonce,
donnera pour la formation de ces fonctions:
[1 2. n] = 1[3 4...] + [4.., 21
+......... +,, [2 3... n- i]."
Dismissing, as not of present interest, the sentence regarding
the generalisation obtained by admitting more than one system
of symbolic numbers, we note first of all the peculiar general
use of (12..n) for any function the expression of which
involves* as suffixes or otherwise the numbers 1, 2, 3,..., in.
Then we are struck with the fact that the use of this along
with ~+ gives a notation for a genus of functions of which
determinants, as understood up to the date of the paper, formed
a species: thus
alb2cS +G a2 + 2cbcc2 -+ 5lb2c1 - - 2b1c - cab3C2
is the case of 2 ~(123), where (123)=acb2%c3. In the third
place we are surprised to find that Cayley seems to propose
to extend the meaning of the word determinant by transferring
the name of the species to the genus, and to call by the name
of "ordinary determinants" the functions formerly known as
"determinants " merely.
All this is in itself comparatively unimportant, serving perhaps
only to recall to us Cauchy's famous paper of 1812, where we
have K, the originating term of an alternating function to
compare and contrast with Cayley's (12... n), and 'alternating
function' to compare and contrast with Cayley's extended
meaning of 'determinant.' But what follows by way of second
example is very noteworthy, because the originating term taken,
viz., X12X34.... Xn_,,, is one that could not possibly have been
used by Cauchy, with whom 2 denoted an operation of a much
less simple character than permutation of the integers 1, 2,..., n.
*Apparently it is meant to be implied that each of the numbers occurs only
once in the expression.
M.D. It. R