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

DETERMINANTS IN GENERAL (SYLVESTER, 1852) 77

a b c. a.., y
a' b c' a'.. a3' y
a" b"c".. by a".. /3" "... 1.. 1I.... 1.. 1...
He thereupon leaves the subject with the remark:"This rule is interesting as exhibiting.... a complete scale whereby
we may descend from the ordinary mode of representing the
product of two determinants to the form.... where the two original
determinants are made to occupy opposite quadrants of a square whose
places in one of the remaining quadrants are left vacant, and shows us
that under one aspect at least this latter form may be regarded as a
matrix bordered by the two given matrices."
The second lemma is the identitya l a 12.... ain 1 Anl A12.... Al 1
a21 %22 a2 2 A2. n 21.2 A2n
anl a2 an,. nl n2...Ann
1 1.... 1. 1 1....1
where A,.= a,., + h,. + c, and the h's and Ic's are perfectly arbitrary
quantities, the transformation being of course effected by adding
multiples of the last column to the other columns, and thereafter
multiples of the last row to the other rows.
The geometrical applications which follow, it may be interesting
to note, are connected with the subject of Cayley's paper of
1841,-his well-known first paper on determinants.
CAUCHY, A. L. (1.853, January).
[Sur les clefs algebriques. Corzptes rendus... Acad. des Sci.
(Paris), xxxvi. pp. 70-75, 129-136; or (Euvres completes (1),
xi. pp. 439-445, xii. pp. 12-20.]
[Sur les avantages que presente, dans un grand nombre de questions, l'emploi des clefs algebriques. Coqnptes rendus...
Acad. des Sci. (Paris), xxxvi. pp. 161-169; or Eivres conpletes (1), xii. pp. 21-30.]
These papers add nothing of algebraic importance to the
contents of Cauchy's memoir of the year 1847: in fact, they may