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

14 HISTORY OF THE THEORY OF DETERMINANTS

and ten of these would have belonged to the 'Theorie.' Only
a few of the additions, however, are new.
One of these is Gordan's mode of verifying the extended multiplication-theorem (~ 4. 8, ~ 5. 3). This will be fully understood
from its application to one of the simplest cases, namely, the case
of the two arrays,
a1 a2 a3 h1 h2 h3
bl b2 b3, 1 k2 k3.
Putting the left-hand member of the desired identity in the form
alh+ a2h2+ a3h3 alk+ a2k2+a3k3 aa a2 a3
blh,+ b2h2+ b3h3 bkl+~ b2k2+ b3k3 bl b2 b3
I. 1.
*.. 1,
he transforms this into. a a3
*. bl b2 b3
- hl - 1.I
-h2 -Jo2. 1.
-h3 -3.. 1,
and then uses Laplace's expansion-theorem to obtain
I alb2 Ih1k2 I + I a1b I I hk3 I + I a2b3 I h2k31,
as Sylvester did in 1852 (Hist., ii. pp. 199-200; also p. 57).
A second result worth noting is a contribution of Kronecker's
(~ 4. 7) to the subject of Sylvester's 'homaloidal law.' Following
on Cayley's work of 1843 (Hist., ii. p. 15), Sylvester had asserted
in 1850 (Hist., ii. p. 51) that by making (n-k 1)2 of the k-line
minors of an n-line determinant vanish all will vanish. What
Kronecker now adds is that the (n-k+1)2 vanishing minors must
have a common non-evanescent primary minor; for example, all the
four-line minors of I alb2cd4e5 will vanish if
I ab23d4 = 0, I ab23d = 0,
Iab2c3e4 = 0, I ab2c3e 1 0,
and [ alb2c3 f 0.