  The Hesse curve or Hesse derivative Hess$(\Gamma_f)$ of a cubic curve
$\Gamma_{f}$ given by a homogeneous polynomial $f$ is the set of points $P$
such that $\det \left(H_f (P)\right)=0$, where $H_f (P)$ is the Hesse matrix of
$f$ evaluated at $P$. Also Hess$(\Gamma_f)$ is again a cubic curve. We show
that for a point $P\in$Hess$(\Gamma_{f})$, all the contact points of tangents
from $P$ to the curves $\Gamma_{f}$ and Hess$(\Gamma_{f})$ are intersection
points of two straight lines $\ell_1^P$ and $\ell_2^P$ (meeting on
Hess$(\Gamma_{f})$) with $\Gamma_{f}$ and Hess$(\Gamma_{f})$, where the product
of $\ell_1^P$ and $\ell_2^P$ is the polar conic of $\Gamma_{f}$ at $P$. The
operator Hess defines an iterative discrete dynamical system on the set of the
cubic curves. We identify the two fixed points of this system, investigate
orbits that end in the fixed points, and discuss the closed orbits of the
dynamical system.
