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

26 HISTORY OF THE THEORY OF DETERMINANTS

data (1) and (4). The entire set is thus consistent, and the initial
array consequently evanescent.*
The freshest of the appendices is the fifth (pp. 128-133), which
deals with the problem of constructing a determinant of monomial
elements which will vanish simultaneously with a given algebraic
expression of two or more terms. Here the fundamental theorem
is that if in an n-line determinant all the elements of a p-by-q array
be multiplied by x, and all the elements of the complementary array
be divided by x, the determinant is multiplied by x'+q-'L. Taking then
the determinant
a b c
d ef
ghi,
he in the first place divides the (2, 2)tl element by itself and multiplies each of the elements of its complementary minor by the same:
secondly, he divides the (3, 3)t1 element and multiplies the elements
of its complementary minor by i: thirdly, he multiplies the 2"d and
3rd rows by b and c respectively: and fourthly, he multiplies the
21d and 3"r columns by d and g respectively. The result of all
this is
a b c aei bdi ceg
d e f = bdi ceg bdi bdi bfg
g h i ceg cdh ceg
where the elements of the determinant on the right are five of the
terms of the determinant on the left. It is consequently suggested
to construct from any six-termed expression
a+O3+y+(8+e+:
* Of course if a 4th column were taken with the additional data
I alb2c4 I = O, I alb2d4 - 0, a1bAe4l = 0,
another 5-by-3 array could be proved evanescent: and still another with the data
Ialbc51 = 0, a,b2,d5,,b2e = I. = O.
We should then have reached a square array, whose 3-by-5 minor arrays we
could treat in the same fashion without additional data, and so prove that if
each of the 9 three-line minors be evanescent, of which the non-evanescent | alb2 is
a common minor, then all the other three-line minors would vanish.