### Properties of a Certain Projectively Defined Two-Parameter Family of Curves on a General Surface

American Journal of Mathematics, Vol. 40, No. 2 (April 1918), 213-234

Excerpt (Section 2)

The theory of two-parameter families of curves on a general surface has received but little attention except in so far as such a general theory may be implied by the theory of geodesics. We shall discuss in this paper a class of curves which will include the geodesics as a special case.

Let us associate with every point P_{y} of the surface one of the lines L_{y} which passes through that point, but does not lie in the tangent plane of the point. All these lines form a congruence L. Let us consider a curve on the surface which has the property that each of its osculating planes passes through the corresponding line of the congruence. All such curves will clearly form a two-parameter family, and it is easy to show that they will be the integral curves of an equation of the form

u''v' — u'v'' + 2(b'^{3} — a'v'^{3}) + 2(p_{1}u'^{2}v' + p_{2}u'v'^{2}) = 0,

where u' = du/dt, u'' = d^{2}/dt^{2}, etc., and where p_{1} and p_{2} are functions of u and v which depend upon the choice of the congruence L.