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

DETERMINANTS IN GENERAL (SYLVESTER, 1852)

71

a y
ax y l\2
(a, b, c,, ex, y)4 = (I) (-4b+3
ax ay
a D
Dx ay
He also uses effectively such umbrae as
ae, as-?1 -3 an2 - -2.
-a usage which is most suitably illustrated by taking a cornmutant having two lines of three quadratic umbrae each, that
is to say, the determinant of the third order
a2 a/3 /32
a2 a/3 /2
This, by a similar convention, is taken to represent
a4. a' 2. a/ - aa3 a.33 3 - / * a3/. /3 + a3/3 a /33 a2. 2
+ a2/32* a3 *. a/33 - a232. a232. a22,
and consequently if
(ax+/3y)4 aC4 +4bx3y + 6cx2y2 + 4dxy3 + ey4,
it is an expression for
ace + 2bcdc- ctd- b2e- c,
that is to say, for the above-mentioned (p. 69) special determinant
a b c
b c d
c d e.
When he comes to speak of 'partial' commutants, which are
identical with intermutants, he devotes a page (pp. 88-89) to the
subject of his relations with Cayley. As it would appear that
he was not quite satisfied with the wording of the postscript
above referred to, Cayley published a modified form of it as a