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

DETERMINANTS IN GENERAL (SYLVESTER, 1851) 59
where, of course, whenever desirable, instead of al, a2,..., a nd
x a2,.., a,, we may write simply a, b,..., 1 and a, /,..., A
respectively. Each quantity is now represented by two letters; the
letters themselves, taken separately, being symbols neither of quantity
nor of operation, but mere umbrse or ideal elements of quantitative
symbols. We have now a means of representing the determinant
above given in a compact form: for this purpose we need but to write
one set of umbrae over the other as follows: (l 2...a, If we
a1 a2 a...,I
now wish to obtain the algebraic value of this determinant, it is only
necessary to take a,, a2,..., a in all its 1.2.3... n different positions,
and we shall have
a1 ag a,\
\{a a2 ~ {}alao01 x a2a2 x., x laO,},
li a2 a,,
in which expression 01, 02,..., 0, represents some order of the
numbers 1, 2,..., n, and the positive or negative sign is to be taken
according to the well-known dichotomous law."
An obvious extension of the notation is also indicated, whereby
what he calls "compound" determinants may be appropriately
represented. Since, in accordance with the foregoing,
a b
is used to denote
aca. b/3 - ct/. ba,
he considers that
a ub cd )
"will naturally denote
ab cdl ab cd
aX - X
a/3 x ~ ya-3,
that is
f (aa x b3) (cy x d8)_ f (ay x b8) (ca x d/3)
- (a/ x ba)J - (x xdy) - (as x by) X - (c/ x da))
And in general the compound determinant
{a ba... *,... a... Or br... 1
al P1i l a2 /2 2... a, r,. r J
will denote
- {{a" b,... 2, X, b2 * * 12 }O x. r I}
I a... J a2 0.. 2j ar. Aor