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

152 HISTORY OF THE THEORY OF DETERMINANTS
interest of the paper is mainly geometrical. Use is made of
the proposition that if the, quadric
ax2 + by2 + cz2 + div2
+ 2lyz + 2inzx + 2nxy + 2pxw + 2qyw + 2rzw
be increased by (atX + o3y + yZ + aqV)2, the discriminant
a n gn.p
n b I q
m 1 c r
pq r~d
is increased by*
a a/ y 3
a a n mn,3 nb 1 q
p q r d
SALMON, G. (1859).
[LESSONS INTRODUCTORY TO THE MODERN HIGHER ALGEBRA....xii + 147 pp. Dublin.]
The evectant (or first evectant) of an invariant I of a quantic
(a, b, e,... x, y, z,..)" being
(DI DIl DaI 7. )
a\a D b'c
*The new discriminant being
a + a2 ni+ca~21 m+cay p-laco
n ~ea3 b+ p I ~fry q + 36
mn-+ ay 1 ~13y C ~ y2 r,+ T&
p +aa q~1369 r+ ya d~+52
is easily seen to be equal to
I - a 13a
a a 9z 92 p
13 n b I q
m In c C
6 p q r- d
and therefore to be separable into the two expressions referred to.