  For general Riemannian foliations, spectral asymptotics of the Laplacian is
studied when the metric on the ambient manifold is blown up in directions
normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is
given in terms of the differentiable spectral sequence of the foliation. The
asymptotics of the corresponding eigenforms also leads to a Hodge theoretic
description of this spectral sequence. This is an extension of results of
Mazzeo-Melrose and R. Forman.
