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

96 HISTORY OF THE THEORY OF DETERMINANTS

regarding the form taken by the evectant of the discriminant of a
quadric when the discriminant vanishes. We see now that the property then stated is equivalent to saying that when the discriminant
of a quadric vanishes, the adjunct quadric is expressible as a square.
Brioschi also notes that if, in addition to the above-mentioned
relation between the a's and A's, there be the following connection
between the x's and $'s, namely,
e1 '- a,1 1 +a+l 2 X... +a.,,,
n a. x ax2..* * + a
so that each $ is the differential-coefficient of the given quadric with
respect to the corresponding x, and consequently so that
Xl1+Xd2 +... +Xn2qn
is equal to the quadric itself, then
AjsgrsC = I alla22, ~ ~ ann I.l'arsXrXs.
It must not be forgotten, however, that the case of this for n equal
to 3 is included in a result noted by Cayley in 1848 (Hist., ii. pp.
116-117), and that the proof there given by us is generally applicable.
As alternative proofs we may now add that
-. -. x y z I..
e a h g f 1... a h g
h b f.. b f
f c 1. f c
a h g
- (xJ+ y+ ) th b...
and that g f
-.x y z 1... = -. /,...
x A IH G a h g xA.
y H B F. h b f y. A
z G F C. f.. A
- (Xt+ 7 + z)A2,
where A stands for the multiplier on the left.
Another fact to be supplied is that Brioschi, when dealing (~ 7)
with the subject of linear transformation, gives a property of pure