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

378 HISTORY OF THE THEORY OF DETERMINANTS

where u33 and u33 stand for ternary cubics, and u33 is the
unknown.
The effect of linear transformation, so strikingly brought to
the front by Boole three years before, is then (~ 19) entered on,
f being no longer a ternary cubic, but any function whatever of
x1, x2..., x,, and supposed to be expressed also as a function
of the variables Y1,,..., y, by means of the equations
(1) (2) (n)
aCC1 + a12 +....+ a, X y
(1) (2) (in)
a2x, + a. x2 + * + a2 x = 2
(1) (2) I) _
a.X, + ax2~ +. an n
Denoting the determinant of f when viewed as a function of the
x's by 95, and when viewed as a function of the y's by O', Hesse
affirms that
r = r2~,
where r is the determinant formed from the coefficients of the
x's in the transforming equations. His proof is essentially that
still followed; that is to say, he recalls that from the multiplication theorem we have
(1) (2) (.1). - a()(2) (2( ) a W,(1),(2). () Wn
I 2...:'= 1~() 2... a, _ 2... w2,
if u() = a()w(l) + a()qw(K) +..+ a( )w(K);
and
~ U1 U/2... )= r - -.V. 2)
if in addition u) = a(x)v(K) + a(X)v ) +.. + a V));
K 1 1 2 2 i IL;
and he then merely asserts that application to the case where
(_) D.2f (K) ( - = a2f
U/K ~ -Qxx ' ~ ~ ' VK ~ y
UK = DXKDXk' ayW-yk
accomplishes the desired aim. The result, as stated in later
phraseology, is that "the Hessian is a covariant." The case
where f= ax2+2bxy + cy2 was given by Lagrange in 1773, and
the case f = ax2 + by2+ cz2 + 2dyz+ 2ezx + 2fxy by Gauss in 1801.