  This paper considers eleven dimensional supergravity on a manifold with
boundary and the theories related to heterotic $M$-theory, in which the matter
is confined to the boundary. New low energy actions and boundary conditions on
supergravity fields are derived. Previous problems with infinite constants in
the action are overcome. The new boundary conditions are shown to be consistent
with supersymmetry, and their role in the ten dimensional reduction and gaugino
condensation is briefly discussed.
