  We show that the addition of points on the tropical Hesse curve can be
realized via the intersection with a tropical line. Then the addition formula
for the tropical Hesse curve is reduced from those for the level-three theta
functions through the ultradiscretization procedure. A tropical analogue of the
Hessian group, the group of linear automorphisms acting on the Hesse pencil, is
also investigated; it is shown that the dihedral group of degree three is the
group of linear automorphisms acting on the tropical Hesse pencil.
