164 LIPKA. 



in the tangent plane to the surface at the given point. The locus of 

 the foci of the osculating parabolas of the associate system is a bi- 

 circular quartic with the given point as node, and the given direction 

 both as tangent line and also as one of the asymptotes to the hyperbola 

 which is the inverse of the quartic with respect to the given point. 



Any triply infinite system of curves on a surface may be represented 

 by a differential equation of the form 



(31) v"'=f(u,v,v',v"). 



Using the same notation and same coordinate system as in §3, the 

 equation of a bicircular quartic described in Property I has the form 



(32) 5o(a^+ ^')-+ B,{a/- ^) (a2+ /3^) + {al''-0) (B.a^ B,0) = 



where the B's are arbitrary functions of v, v, v'. If in (32) we substi- 

 tute for a, P their values as given by (23), we find with the aid of (22) 

 and after considerable reduction, that the differential equation (31) 

 has the form 



(33) v"'= A-^Bv"+Cv"-\ 



where A, B, C, are arbitrary functions of u, v, v'. It is evident that 

 equation (33) is much more general than equation (11). Thus we 

 may state 



Theorem 4. The most general triply infinite system of curves on a 

 surface possessing Property I, is defined by a differential equation of the 

 form (33) involving three arbitrary functions of u, v, v'. 



Let us now con\ert Theorem 3. ^Ye must here replace the direction 

 of the force vector through each point by a fixed direction through 

 each point, but wliich may vary from point to point. "We are to find 

 the most general system of en 3 curves possessing Property I and also 



Property II. The focal locus or bicircular quartic associated with 

 each element (u, r, v') by Property I, is such that the initial element, 

 which has the direction of one of the tangents, bisects the angle be- 

 tween a fixed direction through the initial point and the other tangent. 



Let the fixed direction be given by co (u, v). The system of curves 

 possessing Property I is defined by the differential equation (33). 

 For these curves the focal locus corresponding to the element {u, v, v') 

 has for its second tangent the line 



(34) o\Cv' (1 + r'2) + 3 (1 - /2)] - /3[C(1 + v''") - 6i-'] = 



whose direction is given by 



Ct''(l + r"^)+3(l -v'^-) 



(35) K = 



C(l +/2) -6/ 



