Certain contact properties of linear systems of hypersurfaces

Abstract

where the ('s are homogeneous parameters and thef's are homogeneous functions of degree n in the variables x0, xl, , xr, represents an o P-system, I VI, of hypersurfaces of order n in an r-space, Sr. Interpreting the i's as the homogeneous coordinates of a point of a p-space, 2p, we have a one-to-one correspondence between the points of 2p and the hypersurfaces of VI . In this paper we propose to deal with certain contact properties of the system j Vj and to investigate some of the properties and relations to one another of the corresponding loci in 2p. Our results will be generalizations of certain results which W. L. Edget has, in connection with his study of octadic surfaces, described for the case r = 3, p =2, n = 2. We should mention that T. R. Hollcroft has derived: the properties of the curve in the plane 12 Of the parameters which corresponds to the Jacobian curve of a net of hypersurfaces in Sr, and in another paper? those of the surface in the 3-dimensional space 13 of the parameters which corresponds to the Jacobian surface of a web of surfaces in S3. In neither of these papers, however, has the author touched upon the results which we are going to derive. In the following we shall, first, describe the general case. This is done in ?1. Then we shall consider the case r=p=2 in ?2 and the case n=2, r=p general in ?3. Finally, in ?4, we shall conclude the paper with a description of the results for n = 2, r =p = 3. 1. The general case. Let n, r, p be general. Since for a hypersurface to acquire a conical point or hypernode is equivalent to one condition, there are ooPhypersurfaces of the system I VI each possessing a hypernode. The