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

282 HISTORY OF THE THEORY OF DETERMINANTS
the terms of the aggregate, be it observed) that it is resolvable
into two factors, viz.
( a r s 2/As) s
It is then argued that the two factors are identical even in the
signs of their various terms "da durch das Zeichen einer Wurzel
die Zeichen der iibrigen bestimmt sind"; and that therefore
A = (EcA',
and /A = al, V/A'.,
-an aggregate of n -1 terms, since the values to be given to
r are 2, 3,...,. The next step consists in pointing out that
A',,. being a determinant similar to A but of order n - 2, it must
follow that /A',., can in the same way be expressed as an
aggregate of n-3 terms, and that this process can be continued
until the minor under the root-sign is of the 2nd order, when
manifestly its value is the square of one of its elements. The
final result thus is that //A is expressible as an aggregate
of (n -1) (n -3)... 3.1 terms, each of which is the product of
2In elements whose collected suffixes form a permutation of
1,2,....,.
By way of corollary to this it is pointed out that
~ Ca12c34.. Cn-.,
is one of the terms of the aggregate; and the same is proved by,showing that the square of this is a term of A, the reasoning
being as follows:-Since in every case ars= - as we have
(a12a. a.. c-,) = (a12 34... a1,) (- )(a21a43.. a,-l),.and.'. = (-)f (c12Ca21t34a 43... al,ncn,-l),
which clearly contains n elements, one from every row and one
from every column of A, and will therefore be a term of A if only
we can show that the number of inversions of order in
2,1, 4,3, 6,5,..., nz, -l
is -n, a fact which is self-evident.
21)C C~ VI~I W~Vl Y~IV