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 Py of the surface one of the lines Ly 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(p1u'2v' + p2u'v'2) = 0,
where u' = du/dt, u'' = d2/dt2, etc., and where p1 and p2 are functions of u and v which depend upon the choice of the congruence L.