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

ALTERNANTS (BERG, 1879)17

173

element of the substituted column, th utcm en htten
determinant divided by the old is equal to
la- lb.

For example,
1 a,. (a, - bl)(a,1 —b2)(al —b.)
1 a2 (a2 -b,) (a2- b2) (a2- b3)
1 a3 (a3-b1) (a.3-b2)(a3 —b3)

= 1 a2
1 a3

a13- a~12b +a1lbjb - b1b b3
a2-21 b 2L~2 —b 2b3
a3 -a3 -b +a3 bb2-blb2b3 I

1 a1 a13 1 a1
-'1;1 a2 a23 - I a2
11 a3 a33 1 a3
1 a, a12
1 a2 a2 (c- lb).
1 a3 a32
By similarly substituting in the last column of

a12
a22 Yib
a32

a01a3a4 a0a2a3a4 1a2a3a4
aaa 2aa4ac23t a a a a
we obtain in the same way determinants which when divided by
a 0a~a~a3 1give respectively the quotients
Eaja2a3-Ebbb3
aja2a3a4- b~b2b3b4,
the series of resulting identities degenerating into a well-known
series of early date on making the b's vanish. The already partially
known non-determinant forms of these identities are next given,
being derived by expanding the determinants in terms of the
elements of what we have called the substituted column and their
complementary minors. For example,
(a.,- b1) (a,-b2) (a2-b1)(a2 —b2) ~{a3-b,)(a3-b2) -1
(a.,- a2) (a - a3) ( a2-a,)(a2- a,) (a3- a.,)(a 3-.a2)

(a, -a2) (aj -a3)

+ (a2-b,) (a2-_b) (a2-b3)
(a2-a,)(a2-a1j)
+ (a3 -b1) (a3-b2) (a3 -b3) -,2a -l~b.
(a3-aj)(a3-a2)