New Systems of Hypergeodesics Defined on a Surface

Introduction. Let a non-ruled surface 5 be referred to its asymptotic net as parametric. As a point Py moves along a curve C\ of S, the tangents at Py to the uand ^-asymptotic curves of 5 describe two ruled surfaces R" and R{> respectively. Let Sp and S, denote arbitrary transversal surfaces of the congruences of uand ^-tangents of S, respectively. The purpose of the present paper is to introduce and study systems of curves of 5 which will be called pand a-tangeodesics.