  In this paper we study the behaviour of the Neumann data of Dirichlet
eigenfunctions on simplices. We prove that the $L^2$ norm of the
(semi-classical) Neumann data on each face is equal to $2/n$ times the
$(n-1)$-dimensional volume of the face divided by the volume of the simplex.
This is a generalization of \cite{Chr-tri} to higher dimensions. Again it is
{\it not} an asymptotic, but an exact formula. The proof is by simple
integrations by parts and linear algebra.
  We also consider the following inverse problem: do the {\it norms} of the
Neumann data on a simplex determine a constant coefficient elliptic operator?
The answer is yes in dimension 2 and no in higher dimensions.
