  We consider the Laplace operator with Dirichlet boundary conditions on a
domain in R^d and study the effect that performing a scaling in one direction
has on the eigenvalues and corresponding eigenfunctions as a function of the
scaling parameter around zero. This generalizes our previous results in two
dimensions and, as in that case, allows us to obtain an approximation for
Dirichlet eigenvalues for a large class of domains, under very mild
assumptions. As an application, we derive a three--term asymptotic expansion
for the first eigenvalue of d-dimensional ellipsoids.
