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

CIRCULANTS (PUCHTA, 1877) 389
very concise or clearly worded rule-of-signs,* which is practically
to the effect that The first sign is always -, the second sign is + or
the opposite, the next set of two signs is the same as the first set of two
or the opposite, the next set of four signs is the same as the first set of
four or the opposite, and so on. For example, if the order-number
of the determinant be 24, there are four occasions on which we have
to choose between same (s) or opposite (o), and if the register of
our decisions be s o s s, the resulting factor is
a+b-c- d+e+f-g-h+i+j-k7-l+ m+n -o-p.
What Puchta calls in his title the 'converse' proposition (pp.
220-221) really concerns the finding of a determinant equal to a
product of linear factors which differ only in the signs of their terms.
The illustrative example given, namely,
(a+b-c)(a —b+c)( —a+b+c) = - a-b -c c
-b a-c b
c-b b-c a
is not attractive.
DOSTOR, G. (1877).
[ELEMENTS... DES DETERMINANTS,...xxxi +352. Paris.]
Dostor gives (p. 72) a very peculiar mode of factorizing
a b c d
b a d c
bade
c d a b
d c b a.
This depends on the fact, that if the given determinant be
multiplied by
1 1 1 1
-1 -1 I 1
-1 1-1 1
-1 1 I -1,
* At the basis of this is a proposition which may also be used in practice for
obtaining the factors, namely, If a, be one of the linear factors of the determinant
of the array A, and r, the corresponding factor of the determinant of the array B,
then
A B
B A = (al + 1)(al - 1) (a2 + )(a - 2) (a= g3)(a3 - 3)(a4+P4)(a4- 4)