Surfaces with Orthogonal Loci of the Centers of Geodesic Curvature of an Orthogonal System* by Malcolm Foster

We consider a surface 8 referred to any orthogonal system. Let Ot and G* be the centers of geodesic curvature of the curves u = const, and v = const, respectively, through any point M of 8. As M is displaced over the entire surface the loci of 0± and 02 will in general be two surfaces St and S2, corresponding elements of which are those which result from a common displacement of If. We ask: What are the surfaces 8 for which the surfaces St and S2 correspond with orthogonality of linear elements? The condition that the displacements of C± and C2 be orthogonal for every displacement of M, is that the absolute displacements of these points in the directions of the axes of the moving trihedral at M satisfy the relation