On the Asymptotic Curves of a Surface

hand, conditions sufficient for the truth of the classical statements will be developed in terms which take into account the specifically geometrical origin of the various differential equations defining the respective problems. Oii the other hand, examples will show that the conditions to be imposed cannot be omitted. Both an attraction and a difficulty of the theory is the circumstance that, even if the surface has a parametrization in terms of asymptotic curves, examples show that this parametrization of the surface can be less differentiable than some other parametrization of the surface (or, equivalently, less smooth than the parametrization z = z (x, y), where x, y, z are Cartesian coordinates). Surfaces of negative Gaussian curvature are dealt with in Parts I, II, III. Part IV deals with lines of curvature and applies therefore to surfaces of positive curvature also. Finally, Part V develops the corresponding theory