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

124 HISTORY OF THE THEORY OF DETERMINANTS

The " demonstration " leaves a good deal to be desired. In effect
it amounts to saying that if 1,,..., p be the pth roots of
unity, and D,n the determinant got from 1(11) (22)... (nn) by
subtracting ~,,x from each of the diagonal elements, then
D1D2D3... Dp [11]-XP [12].... [1n]
[2 1] [22] -... [2n]
[nl] [n2].... [nn]- -
Now it is well known that the multiplication of D1, D2,..., Dp
enables us to arrive at the equation whose roots are the pth powers
in question, but this and Sylvester's statement are not by any
means identical. The separate points to be established are
(1) that the element in the rt" row and sth column of the
determinant which is the product of D1, D2,..., Dp consists
of [rs] and a tail of other terms, (2) that this tail vanishes in
the case of every non-diagonal element, (3) that in the case of
the diagonal elements it reduces to - x; and Sylvester's only
justificatory statements are that the product of the p determinants
is independent of the order in which they are taken, and that
all the terms containing a ~ in any other power than the pth will
vanish.
Another true proposition made on insufficient foundation is
that the pth power of an axisymmetric determinant is itself
a(xisymmetric. The foundation here is the incorrect proposition
of ~ 6.
CAYLEY, A. (1852, Dec.).
[On the rationalisation of certain algebraical questions. Cambridge
and Dub. Math. Journ., viii. pp. 97-101; or Collected Math.
Papers, ii. pp. 40-44.]
The equations first considered are of the type
a2 +b2 +c- +.... = 0,
and the fresh departure consists in viewing such an equation as
the outcome of the set of equations
x+y+z+... = 0, 2=a, y2=b, =c,...