PROPERTIES OF THE HESSIAN 
25 
§ 13] 
under projective transformation. If the triangle of reference 
is equilateral and the coordinates are proportional to the per 
pendiculars upon its sides, X1X2 — x 3 2 = 0 is a circle (§ 12). 
On the contrary, if we employ only translations and rota 
tions, as in plane analytics, there are infinitely many non 
equivalent conics ; we saw in § 1 that there are then two 
invariants besides the discriminant. 
Next, to make an application to plane cubic curves, let 
f(x 1, x 2 , x 3 ) be a ternary cubic form. A triangle of reference 
can be chosen so that P=(00l) is a point of the curve/=0. 
Then the term in x 3 3 is lacking, so that 
/ = X 3 2 fi + X3/2 +/3, 
where ft is a homogeneous function of x\ and x 2 of degree i. 
We assume that P is not a singular point, so that the partial 
derivatives of / with respect to xi, x 2 , and x 3 are not all zero 
at P. Hence /1 is not identically zero and can be introduced 
as a new variable in place of x\. Thus, after a preliminary 
linear transformation, we have 
x 3 2 x\ -\-x 3 {ax\ 2 +Ъх\х 2 -\-cx 2 2 ) +/3. 
Replace x 3 by x 3 —|(axi + bx 2 ). We get 
F = x 3 2 xi + ex 3 x 2 2 +C, 
where C is a cubic function of xi, x 2 , whose second partial 
derivative with respect to Xi and Xj will be denoted by Cy. 
The Hessian of F is 
Си C12 
H = C12 C 22 -\-2ex 3 
2x 3 2ex 2 
2хз 
2ex 2 
2xi 
If the transformation which replaced / by F is of deter 
minant A, it replaces the Hessian h of / by H = A 2 h. Thus 
H = 0 represents the same curve as h = 0, but referred to the 
same new triangle of reference as F = 0. We may therefore 
speak of a definite Hessian curve of the given curve /=0. 
In investigating the properties of these curves we may therefore