  We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and
prove some general result concerning the linear system $|H-P|$. We then look at
regular surfaces lying on hypersurfaces of degree $s$ having a plane of
multiplicity $(s-2)$. This implies that $S$ contains a plane curve. We prove
that the degree of such surfaces is bounded and for $s=4$ we compute an actual
bound.
