Degenerate surfaces in higher space

Abstract

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).