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

COMPOUND DETERMINANTS (KRONECKER, 1869) 1191

KRONECKER, L. (1869).
[Bemerkungen zuir Determinanten-Theorie. Crelle's Journ., lxxii.
pp. 152-175; or, Werke, i. pp. 237-269.]
The theorem which the first of the notes concerns is ~ 4. 7 of'
Baltzer's second edition (see above, p. 14), and is to the effect that
if the determinant Ialla22.... am,, j be bordered in n 2 different wvays,
the typical resulting determinant being
a,, a12.... a,,,, alk
a1 a2.... a19 ak
or Bil, say,
a11 a19.... a1,a,
where li and k are any one of the first n integers and n > m, then all
the (m + 1) -line minors of I B1,iI must vanish. The proof consists
in expressing each B in terms of the elements. of its last row and
their complementary minors, and then pointing out that any one
of the (m +1)-line minors in question, as thus altered, can be seen
to be expressible as the product of two vanishing determinants.
Hesse's compound determinant like I B,1, I may be worth recalling
(Hist., ii. pp. 130-132).
HUNYADY, E. (1875, 1876).
[A k'LIpszeleten fekv6 hat pont felte'teli egayenlet~nek k~l6nb~iz6
alakjairo'l. Ertekeze'sek a math. tudoaminyok kore'bol (Budapest),
iv. 6 (23 pp.)]
[Ueber die verschiedenen Formen 'der Bedingungsgleichung, welche
ausdrtickt, dass sechs Punkte auf einem Kegelschnitte liegen.
Crelle's Journ., lxxxiii. pp. 76-85.]
What concerns us here is a three-line determinant which is doubly
compound, that is to, say, a determinant whose elements are themselves compound determinants. If we put
X,k for y11z1,
Y1,, for I~, and I Y,,~,Z,,,,W for Y>Z,,kz1,, - 1MIU
Zh 1,, for xh;yI J