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

ORTHOGONANTS (BRIOSCHI, 1854)

317

element. The value of this is shown (p. 65) to be 0 when n is
odd, and 2"^A/A when n is even, A being the basic determinant,
and A0 what A becomes on making all its diagonal elements zero.
The result is easily reached on multiplying the given determinant
by A and showing that the product is (-l1)n2"A0.
BRIOSCHI, F. (1854, August).
[Note sur un theoreme relatif aux determinants gauches. Journ.
(de Liouville) de Math., xix. pp. 253-256; or in the French
translation of his Teorica dei Determinanti, pp. 144-147;
or Opere mat., v. pp. 161-164.]
Brioschi's subject is really the equation
(C11 - X (012.. ln
W~ -X Wo... (o -
C21 (1)22 X.. C. ' 2n - n
Wnl WLn2... Wmn, -
in which the left-hand member is the determinant of Cayley's
orthogonal substitution with - x affixed to each diagonal element.
He notes at once, of course, that if the basic determinant be
1 a 22... a,, 1, or A say, the equation may be changed into
All -y A12. An
A21 A22-y... n = 0,
Anl An2... Ann-y
where y is put for (1 +x)A. A further transformation is then
effected by multiplying both sides by A and putting z for
1- A/y, the result being
z a621... Cl
a12 z.... a2 0.
C an. 2..
Using Cayley's expansion (1847) for the determinant on the left,
it is seen that when n is odd the equation resolves itself into
z=0 and an equation in z2 with positive coefficients, and that