The Hypersurface Cross Ratio

Introduction. In my note on A 5 curve theorem generalizing the theorem of Carnot, I introduced the notion of curve cross ratio. This extension of the ordinary cross ratio is the simplest situationally invariant (3.4) case of the generalized or hypersurface cross ratio of n + l pairs of hypersurfaces in w-space which is the subject of the following lines. The generalized cross ratio is a t the same time a generalization of the resultant of n + l quantics; the connection between cross ratios and resultants occurred to me when reading a paper of P. Humbert. The properties of the generalized cross ratio, including extensions of some of those of the ordinary cross ratio, will be developed, together with the similar and interdependent properties of an analogous generalization of the intersection of n hypersurfaces to pairs of hypersurfaces, in §3. This section, much of the contents of which is known, is parallel to §§1 and 2 on the ordinary resultant and intersection. In each section, after the definition and fundamental properties, the influence of a rational transformation of coordinates and of permutation, variation and linear combination of the hypersurfaces is studied.