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

SKEW DETERMINANTS (BRIOSCHI, 1854)

27.5

being used conventionally to denote a certain minor of P, and
the differentiation on the right being real. By squaring we have
(DP) p ( 21)2
and since, as we have seen, it is permissible to substitute
P,( for (
DCtaaDss \odars
there results
(32 p \2 otH
Ctrr ~ts)= -+rs
so that the expansion for P above obtained may be altered into
P S(E ars H) 2
from which by extraction of the square root we have
H (= ars 3_
This will be recognised as a third mode of writing an already
well-known result, and, as Brioschi notes, gives a property of
the function H similar to a property of determinants (" la quale
equazione contiene una proprieta della funzione H analoga ad
una nota dei determinanti ").
From this he passes to what he calls the characteristic
property of H, viz., its change of sign consequent upon the
transposition of two indices. Calling H' what H becomes when
r and s are interchanged, he notes that in those terms of H in
which the element as occurs there can be no other element with
the same indices, and that therefore
aH - aH
aCtrs Dtrs
Then since the same interchange made in P leaves P in reality
unaltered,-that is to say, since H2= H'2,-he obtains
- H - H'
H =11'
DUars aar,