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

32 HISTORY OF THE THEORY OF DETERMINANTS

A A A2x A As A 2 x
+ As2 Yl BB + B B 3+ B2 B+ 2 Y
Z1 C2 C3 C1 Z2 C3 C+ C23 J
-s3 A1B2C3 = 0;
and, by equating like powers of s in the two, three results are
reached, namely,
*I V2XW3 1 I XI123 1,
*AIB2
f X3 A, X A, aXg A_ x? 3
A12C{Y1 $2 3 1 A2 3 A1 2 x3
IvW A2 B Y B +x B, xa
V2W3 +litW31 + lV21 = Y1 Y2 3 + Y B1 23 + B12 3:
I Al.-, B2931 x %1 -2z z2 C3 z1 C2 Z3 C1 z2z:
X1 A A A A 1 A A2 X3
1 + V2 + W A3 = A2C3 i B2 B3 3 B1 Y B3+ 1 B2 Y
zi C2 03 Cl z2 C3 C1 C2 3
In the first of these the cofactor of Ixly2z3 is independent of
the x's, y's, z's, so that it does not alter on putting
( $1 (1z 1. )
X2 Y2 Z2 =
X3 Y3 3..31;
and as on doing so we have
all 21 2[X1 C3x31
bly2 by22 322 = b AByC X 2|
c1z3 c2 3 c3Z3
it is seen that the said cofactor is equal to A. This is Cauchy's
old theorem regarding the adjugate determinant, and substituting
A for the cofactor, we have another equally old, namely, the
multiplication-theorem.
The reader should note, however, that it is better not to view
these old results as the goal of the memoir; but, viewing them
as already known, to note the theorem then reached regarding
the sum of the m-line coaxial minors of the product-determinant.