Multiple tangents to forms in four dimensions
Leonard Roth · Mathematical Proceedings of the Cambridge Philosophical Society · 1930
Formulae for multiple tangents to the general surface in ordinary space were obtained at different times by various writers, but the discussion remained incomplete until the advent of Schubert's enumerative method, which solves the whole problem by a purely mechanical process. Schubert later extended the method to general forms in [n], in the restricted case where the multiple tangents have only a single contact. In the present paper the results for three dimensions have been used to build up the formulae for forms in [4], by means of the correspondence theorem on which much of Schubert's work is based. It would no doubt be possible to evolve a complete set of incidence formulae for four dimensions and then to proceed as in Schubert's discussion of surfaces; but the present method is preferable for two reasons. In the first place, all the results have been obtained in a very simple manner; and secondly, a large number of minor results have been found in the process.