SURFACE MOTION FOR -ANY POSITIONAL FIELD OF FORCE. 167 



hyperosculating trajectories which pass through any point on the surface, 

 is a conic passing throiigh the point in the direction of the force vector. 

 We shall call this conic the central locus. 



§ 6. Curves with Properties I, II, and III. 



We shall now find all triply infinite systems of curves on a surface 

 possessing Property I and the following 



Property III. Through every point on the surface and in every 

 direction through that point there passes one curve of the system 

 which hyperosculates its corresponding geodesic circle of curvature. 

 The locus of the centers of geodesic curvature of the oo^ h3T)eroscu- 

 lating trajectories which pass through a point is a conic passing through 

 the point in a fixed direction. 



To find all such systems we shall evidently have to convert Theorems 

 2, 6, and 7, replacing the direction of the force vector at any point by a 

 fixed direction co (w, v) through that point. By Theorem 4, a triply 

 infinite system of curves possessing Property I may be represented by 

 a differential equation of the form 



(33) 'd"'= A + Bv"+Cv"\ 



The condition for hyperosculation of the curve and the element 



{u, V, v') is given by 



(42) (1 + v' -)G'- 6'{ (X„+ xy) (1 + / 2) _j_ 3^v'} = 0, 



where G and G' are given by (9) and v'" is to be replaced by its value 

 from (33). Now G = must satisfy (42), suice the geodesic with 

 element {u, v, v') certainly hjT^erosculates (indeed coincides with) its 

 corresponding geodesic circle of curvature. And if, excluding the 

 geodesic, there is to be only one curve of the system which has the 

 hyperosculation property, equation (42) must give only one value of 

 v". Now G is Imear in V', and G' is quadratic in v", so that G' must 

 have the form 



(47) G'= Gia + hv") 



where a and h are functions of u, v, v' only. Equation (47) is therefore 

 a restriction on the forms of the quantities A, B, C appearing in (33). 

 Introducing (47) mto (42), we get for the value of v" corresponding to 

 the direction of hyperosculation, 



[a- (X„+Xy)][l+/-] 



.jf 



<*^^ "- 3/-t(l+.-) 



