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

SYMMETRIC DETERMINANTS (SCOTT, 1879)

131

The result of bordering the given determinants, by prefixing
horizontally and vertically

to the first, and

0, 1, 1, 1,
0, a, b, el.,

to the second, is a pair of determinants with the values
e (x -2ao) (x-2b)(x-2c... { + - +x-2~a X_ 2b + +2
a2 b2 el2
- x"-1* (x-2a)(x-2b)(x-2c).. { a + 2b + x2c +
The case of the former value where
x = a-b+c+.
is fully worked out for n 3 and for n = 4.
In his text-book, which appeared the following year, he goes a
little further (pp. 216-218), bordering axisymmetrically the first
of the initial determinants above with the two lines
1, 1,...,1, 0, 0,
) $2' * * 2 93,, 0, 0,

the evaluation being
X*-1. f —2)(x-2a)x2)..

I 1
x-2at x9b~b
_ 1 $ 2 +
x-2a x-2b

01 + i8
x-2a x-2b
$2 +2
x_ $2 x 2b
x —2a z-2

and generalizing the second initial determinant so as to have
Xi a2b1 a3b1 * = {(Xj-a,bj)(X22-ab2)..
aGb3 abb3 X 1+ a
a1b X2 a2 + *albi a~b
a-,b.,a~b,, X.13... X-alk + X2 —a~b2
He also notes (p. 214) the peculiar 2n-line determinant whose odd
rows have an a where they intersect the diagonal and a b elsewhere,
and whose even rows have the reverse construction, the evaluation
being
* (n-1) - (a-b)n.