SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 165 



For the bisection property we must have 



Oi — v' v' — K 



1 -\- (jOV 1 -\- KV 



from which we find 



(37) C = -^ 



V — CO 



SO that C is no longer an arbitrary function of u, v, v'. We may now 

 state 



Theorem 5. The most general triply infinite system of curves on a 

 surface possessing Properties I and II is defined by a differential equation 

 of the form 



(38) v'''= A+Bv"+-y^v"\ 



V — 00 



involinng two arbitrary functions A and B of u, v, v', and one arbitrary 

 function CO of u, v. 



Since properties I and II do not characterize our system of tra- 

 jectories we shall seek further properties. 



§ 5. Hyperosculation and the Central Locus. 



Consider a point on the surface. The geodesic curvature and 

 center of geodesic curvature of a curve through are respectively the 

 curvature and center of curvature of the orthogonal projection of the 

 curve on the tangent plane to the surface at 0. Using the same 

 codrdinate system as in §4, we fuid that the geodesic curvature is 



1 _ G ^ v" - {\,-Kv') (1 -{- v' '-) 



^'^ ^ p~eMl+/^)^ e^a+v'^Y- 



and that the coordinates of the center of geodesic curvature are 

 (40) ^ = - -7=7^' ^ = 



Vl+v"'' Vl+v'^ 



Now, for each trajectory c which passes through we may draw the 

 curve g which osculates c and which has constant geodesic curvature 

 (that of c at 0) throughout. We call g the oscidating geodesic circle of c. 

 The question arises: how many of the oo^ trajectories which pass 

 through in a given direction v', mil hyperosculate (have 4-point 



