Projective differential geometry of curved surfaces

Abstract

The theory of surfaces, as developed by the author in the preceding three memoirs, was based upon the assumption that the surface considered was referred to its asymptotic curves. The complete system of invariants and covariallts was set up on that hypothesis. In order to make the theory directly applicable to a surface whose asymptotic lines are not known, as well as for the sake of formal generality, it becomes necessary to develop the theory of invariants and covariants of a system of two linear homogeneous partial differential equations of the second order with one dependent and two independent variables in its most general form. It is the purpose of the present paper to develop the essentials of this theory. The calculations, some of which are very long, will only be indicated. The final expressions for the invariants, however, are as simple and elegant as can be desired.