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

78 HISTOIRY OF THE THEORY OF DETERMINANTS

BATTELLI, S. (1878-9): LIT, R. R. (1879).
[I PRIMI ELEMENTI DELLA TEORIA DEi DETERMINANTI, con alcune
applicazioni all' algebra ed alla geometria. ~ Sch. Progr. Rovereto.]
[BEGINSELEN VAN DE LEER DER DETERMINANTEN, met voorbeelden
entoepassingen. viii+38pp. Amsterdam.]
Both of these are quite elementary, and there is nothing about
them more than their titles would lead one to expect.
SIMONNET (1879).
[Sur les conditions de l'existence d'un nombre determine de
racines communes ' deux equations donnees. Comptes Rendus. Accad. des Sci. (Paris), lxxxviii. pp. 223-224.]
The basis of the special investigation here is a carefully formulated theorem in general determinants which we may put in the form
I aj17jB,,. = A Ib ijb (1.),1 + A2.29.1 bpnj (k),2 4 - - 4 + A7mrlZ IblI (k),n
where I a,,,,, IblIare any two determinants; Aim f, jB,,,
their adjugates; and I bl,, 1,, what I bf,, becomes when its kth,
row is supplanted by the first n elements of the sth row of I al. I.
No proof is adduced; but if, in order to gain breadth, we view the
theorem as giving the product of two determinants of the pth1 and
qth orders (pr> q) in the form of a sum of products of two determinants of the (p l)th and (qJl)t11 orders, there is no difficulty
in establishing it, and its relation to previous work becomes
more readily apparent. For example, when p, q - 4, 2 (or, when
m, n = 4, 3), we have
ab2c3d4J I elf9= 2 a, a2 a3 a4. l= a a2 a3 a4 - a.1 -a
b, b2 b b3.4 b, b2 b3 b4 -b, -b,2
C1 C2 C3 C4 C c2 c3 C4 -C1 -C2
dl d2 13 d14. dl d2 13 (4 -d1 -d2
e1 e2 e3 e1 e2 el e2 e3
11 f2 fA. A f2,f A f 3 f.
aI 1b2c4 I'I dle2J3 I +-I alb2d4 I-I Cle2f3 I
-Ja1C2d4 H,1 ble2hf I + Ibcd IbIa edf3l

In Sylvester's similar theorem of 1839 p and q are equal.