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

SYMMETRIC DETERMINANTS (FERRERS, 1866) 125
(b + c)2 b2 C
a2 (ea eCa 2 2 = 2abc(a+be),
2 b2 (a b)2
given the same year in Salmon's Modern Higher Algebra (p. 14).
The determinant has the simple axisymmetric form
(b c)2 ab aLC
ab (c+a)2 be
ac be (a+b)2
SARDI, C. (1868).
[Un teorema su' determinanti. Giornale di Mat., vi. pp. 357-360.]
Sardi viewed his determinant as a generalisation of Torelli's
axisymmetric determinant of 1864. When of the fourth order it is
x1 a2 a3 a4
a1 x2 a. a4
al a2 x3 a4
al1 a2 a3 x4
all the elements of any column being identical except the diagonal
element. Of his two w'ays of treating it the second is the better,
namely, changing it into a determinant of the fifth orderby appending
a row of four zeros and then a column of five l's. This five-line
determinant is then seen to be equal to
xi - al I.
* X2Ct..2
x* - a7L *
X4 a4
-al -a2 -a. - 1
rand therefore equal to,- a1)(xX2-a2)(X- a3) (x4- a4)
~ (x1-a,)(x,-a2)(x. —a3) a.
~ (x1-a,)(x2-a2)(X4- a4) a.
+ (xl-a,)(x3-a3) (x41-a4). a2
~ (x2-a,)(x3-a.)(x4-a4) a,