If two projective pencils of curves of orders n and have no common curve, the locus of the intersections of corresponding curves of the two is a curve of order through all the centers of either pencil. Conversely, if a curve of order contains all centers of a pencil of order n to the multiplicity demanded by Noether's fundamental theorem,then it is the locus of the intersections of corresponding curves of this pencil and one of order projective therewith.