624 
MATHEMATICS: J. LIPKA 
proc. n. a. a 
II'. For an "w" system, the focal locus or bicircular quartic associated 
by property I' with each direction ^ through a point 0, is such that the 
tangent of the angle which the tangent line to the quartic in the direction 
^ makes with the force vector is to the tangent of the angle which the 
second tangent line to the quartic makes with the first tangent line as 
w + 1 is to 3. 
III'. The "w" system possesses property III of section 2. The conic 
described in this property is a rectangular hyperbola. 
IV'. For an "/i" system, at any point O on the surface, the geodesic 
curvature of the line of force through 0 is equal to (n + 1) times the 
geodesic curvature of the hyperosculating curve of property III' which 
passes through 0 in the same direction. 
v. For an "w" system, as we move along the surface from a point O, 
the quantities pi, p2, Ri, R2, w, ^i, S2, defined in property V of section 2, 
vary so as to satisfy the relation 
where 
Ki \pi Ki / K2 CO \p2 Ko / 
If, in these five properties, we replace the direction of the force vector 
through a point by the direction of the tangent line to the rectangular hy- 
perbola of property III', we may state the theorem: 
In order that a triply infinite system of curves on a surface may he identified 
with an "w" system, the given system must possess properties I', W, IIP, 
iv\ v. 
^ This is a summary of a long memoir offered for publication to the Ann. Math., 
Princeton, N. J. For a study of the corresponding problem in a plane or in ordinary 
3-space, see Kasner, E., Trans. Amer. Math. Soc, New York, 7, 1906 (401-424); also 
Ihid., 8, 1907 (135-158). For a summary of these two papers, see Princeton Colloquium 
Lectures, Chap. I. 
2 Dots refer to total derivatives with respect to the time t, and primes refer to total 
derivatives with respect to the parameter u. 
3 For a characterization of the curves of a natural system on a surface, defined 
by fPds = minimum, where 7^ is a function of the coordinates only, see Lipka, 
J., Ann. Math., Princeton, N. J., 15, 1913 (71-77). 
