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

156 HISTORY OF THE THEORY OF DETERMINANTS

we have
Ea2b2 = alb2c3dP - a0b2C3d5 I + a0blc4d 5
aOblC2d3 a biC2d3 a0b'c2d3 C '
and therefore, by Naegelsbach's theorem,
4 C03 C3 02
The accuracy of this is readily verified, for the right-hand member
= 2C4-2C1C3 +022
= 2abcd-2 (12a2bc+4abcd) +(1,a2b2 +212a2bc+6abcd)
- Ca2b2.
The rest of the interesting paper deals with certain special cases
and applications.
SCHRODER, E. (1875).
[Ueber v. Staudt's Rechnung mit Wflrfen und verwandte Processe.
~ 6. Math. Annalen, x. pp. 297-301.]
Schr6der's result is implicitly included in Beltrami's, from which
it can be deduced by putting one of the given variables equal to 0.
Thus, putting v = 0 in the example given under Beltrami, we obtain
a1+a2X -a3X2 a%+ay4+a3,x2 a0
b 2+b2X b3X2 bb~b2~x+b31U2 b2 +
c1+ C2X C1 +C2,A C
ao a, a2 a3
bo b1 b2 b3
Co Ci C2
V1 -X-JA I
which is an example of Schrider's.
VALERIANO, V. (1876).
[Alcune notevoli applicazioni della induzione matematica. ~ 6.
Giornale di Mat., xv. pp. 41-45.]
The subject is the factorizing of
0 r 1 n-1 a2... n-2n
Ia aa, a> I and fa -,aa I
Brioschi (1854) might have been referred to.