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

80 HISTORY OF THE THEORY OF DETERMINANTS

which, by Laplace's expansion-theorem, is equal to
I acdb2 'c3d41 c e - \ajb3c4d51 * 1c2d3e41 + a2b3c4dl I cd3e4l.
Next we perform on the determinant the operations
ol13+col5, col4+col6
row5 -row3, rowg -row4,
and so obtain the form
al (a2 a 4.. a,5
b, b2 b3 b4
C1 C2 C3 c4 C5
dl d2 d d4.. d... C4... d3
e, e2 e3 e4 e3 e4 e5
which, by again using Laplace's expansion-theorem, is seen to be
equal to
I b2c3d4e5 I I d I
The required result is thus reached.
Of the second theorem the first case is identical with that given
by Hermite in 1849, and the other cases result from the continuance of the condensation so begun. For example, n being 5, we
have
a. a2a a3 (4 a'5 I= lalb2I l a2b31 a36b4 I a4b51 - a2a3a4
b, b2 b3 b4 b5 a1c21 I a2c3 I ac4 I I a451
c1 C2 C3 C4 c5 a1d2 1 a2d1 3 a3d4 | 1 a4d5
d d(12 d3 d4 d5 | a |e2 I a2e31 I a3e41 Ia4e51 I
el e2 e3 e4 e5
Ia|lb2c3 a2b3c4 I a3b4c5 -. a2b31 * a3b41
alb2d3 I a2b63d4 | a3b4d5
ab2e3 I J a2b3e4 I I a3b4e5 1
| I ab2c3d4 I a2b3c4d51 I a2b3c4.
I ajb2c3e4 a2b3c4e