170 LIPKA. 



Comparing (58) and (59), we may state 



Theorem 10. For (my point on the surface, the geodesic curvature of 

 the line of force is equal to three times the geodesic curvature' of the hyper- 

 osculating trajectory which passes out in the direction of the force vector. 



Let us now find the s^^stems possessing Properties I, II, III, with the 

 additional property got by converting Theorem 10. We replace the 

 force vector by the fixed direction co, tangent to the central locus of 

 Property III. We may now ask for all the triply infinite systems of 

 curves on a surface which possess Properties I, II, III, and 



Property IV. With each point on the surface 0, Property III 

 associates a direction through the point, viz., the tangent to the 

 central locus or conic. The totality of all such directions on the 

 surface, defines a simple system of coi curves, which may be called 

 the tangential lines. The geodesic curvature of the tangential line 

 through is equal to three times the geodesic curvature of the hyper- 

 osculating trajectory which passes through in the same direction. 



The geodesic curvature of the tangential line v' = o^ through is 



p e (1 + w")- 



The curves possessing Properties I, II, III are defined by the differ- 

 ential equation (56). For the hyperosculating trajectory in the 

 direction v' = co, we have, by (55). 



// ^ To + 7i^ + 72*^" 

 3 

 and for the geodesic curvature, 



/p.N ^ ^ (to + Ti<^ + 72C0^) — 3(X„ — coXj (1 + CO") 



^ ' R~ Se^l+co^)^ 



Setting 1/p equal to three times 1/R, we find 



(62) 7o + 7i<^ + y-ior = (cou + wcoi,) + 2 (X„ — ojXJ (1 -f- co~). 



Hence we have 



Theorem 11. The most general triply infinite system of curves pos- 

 sessing Properties I, II, III, IV, is defined by a differential equation of 

 the form {56) together ivith the condition {62), thus involving three arbi- 

 trary functions of u, v. 



We must therefore seek one other geometric property which would 

 reduce the number of arbitrary functions of u, v in (56) to two and thus 

 reduce this equation to that of the trajectories (11) ; this fifth property 

 would then complete the characterization.' 



