  We show that the gradient and the hessian of the Riemann theta function in
dimension n can be combined to give a theta function of order n+1 and modular
weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus
of this theta function essentially gives the ramification locus of the Gauss
map. For jacobians this leads to a description in terms of theta functions and
their derivatives of the Weierstrass point locus on the associated Riemann
surface.
