On the Affine Classification of Quadric Loci

The classification of plane conics into hyperbolas, parabolas and ellipses can, it is well known, be extended to Euclidean-affine space of any number of dimensions. It is the purpose of this note to shew that the classification can be completely described by four numbers which are respectively the rank and signature of the quadric itself, and the rank and signature of its section by the prime at infinity. These four numbers are independent in the sense that no one of them can be unambiguously determined from the others. The description of quadric-types by the values of these four numbers is very convenient, and does not appear to have been stated before. The usual terms, ellipse, hyperbola, parabola, cylinder are used in an extended sense, and serve to characterise affine types.