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

476 HISTORY OF THE THEORY OF DETERMINANTS

HORNER, J. (1865).
ENotes on determinants. Quart. Journ. of Math., viii. pp. 157-162.]
As an example of his theorem regarding the reduction of the order
of a determinant, Homner gives
b C
1. d — e ---
i d e a a
a
I I~~~a
b f! Id b f b
I I i1 aI b
cill ecf C
Cfl
ad-b ae-c abd cef
b -ad. (bf- c)d
(c-ae)f (c-bf)e
(b - ad) (c - bf) (ae - c) (e - df) +- abedef.

HORNER, J. (1865).
(Notes on determinants. Quart. Journ. of Math., viii. pp. 157-162.]
Instead of using { } in the symbolizing of a permanent, as Cayley
did in 1857, Horner introduces I and gives the identity

= ad be ef
ald1 be, clf1
a2d'2 b2e2 42f2
- ad1 be1 cJ1
aCd be clf
a2d2 bVe2 c2f2

(t b I d e
ati b c,1. ld, e1
(1 2 b2 c2 (12 e2
- ad be 0af
a,1d2 b1e2 e~f2
a2c1, b2el e2f1
-- ad2 be2 cfA
ald ble c~f
a2d1 b2el c2fA

f
fi

~ ad, be, CA1
a1d(2 b~e2 cA A
Ia2d b2e c2f
ad2 be2 Af2
a"d1 b e1 c(I 1f
a~d b2e c2f

as an instance of a general theorem which he establishes for expressing the product of two determinants in the form of an aggregate
of what he calls 'conterminants.' His proof depends on the fact