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

276 HISTORY OF THE THEORY OF DETERMINANTS

and, it having been shown that the two differential-quotients
here appearing are of opposite signs, it follows that so also are
H and H'.
Lastly, he passes on to skew determinants in general; and,
using the theorem and notation introduced at the outset, he
writes Cayley's propositions in the formn even, P = PO + Er EsArass(2pii)o +...+ a11a22.*.' Lnnl
n odd, P = rC,rr(19ii)o +.. + lla22.' ann,
which, he says, when the principal elements are all unity, become
n even, P = PO + i(2,ii)o +... + 1,
n odd, P = ^i(pi)0o + Pi(3p)o +.. + 1,
the development now being in each case a sum of squares, as
all the minors appearing in it are even-ordered.
BRIOSCHI, F. (1855, March).
[Sur l'analogie entre une classe de determinants d'ordre pair et
les determinants binaires. Crelle's Journ., lii. pp. 133-141;
or Opere mat., v. pp. 511-520. See also Anrnali di Sci.
mat. e fis., vi. pp. 430-432.]
After explaining that his purpose is to generalise a result of
Hermite's (Comptes rendrts.. Acad. des Sci., Paris, xl. pp.
249-254) regarding determinants of the fourth order, Brioschi
sets out by establishing a necessary lemma regarding determinants of any even order whatever. It is this lemma which is
of importance to us in the present connection. Taking the
determinant
E( al11 22.... a62m, l, 2m or A say,
he multiplies it by the equivalent determinant
a12 -1l C14 - a13.... al,2, 2nz a,2m-1
a22 - a21 a24 - a23.... L2,2m - a2, 2i, -
a2m, 2 a2m, 1 a2mn, 4 -~ C2m, 3 *. * 2m, 2 2m, 2m- 1