SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 179 



We further find that through every point and in every direction 

 through that point there passes one trajectory which hyperosculates 

 its corresponding geodesic circle of curvature. This trajectory is 

 given by 



(87) '" 



1+/2 



(1+70(0 + '/'^'') 



and the central locus, or locus of centers of geodesic curvature of the 

 00 "^ hyperosculating trajectories which pass through any point on the 

 surface, is a conic passing through the given point in the direction of 

 the force vector. The equation of this conic is 



(88) e (0. - X.0 - K^P) + ^v i^v - ct>u + 2X,0 - 2\,.^p) 



- fi^u - \oP - X.0) + (1 + n)c^ {(t>ri - yp^) = 0. 



Since, by (81), 0„ = -^u, this conic is a rectangular hyperbola. Hence 

 Theorem 18. The " n " system possesses Property III. The conic 



described in this property is a rectangular hyperbola. 



We may now show that the most general triply infinite system of 



curves possessing Properties I, IF, III, has an equation of the form 



(89) (co - /) 6" = 6' I TO + Ta / + 72 / ^ + [(2 - n) |±^^ - 3V I , 



where 70, Ti, 72, w are arbitrary functions of u, v. 



From (88) we conclude that at any point the asymptotes of the 

 rectangular hyperbolas associated with all "71" systems are parallel, 

 and that the locus of the centers of these hyperbolas is a straight line 

 through 0. 



We further find that the geodesic curvature of the hyperosculating 

 curve (87) which passes out in the direction of the force vector \p/(f) is 



.QAN 1 _ 0V« - ^v. + ct^^p i^v - 0u) - (X.0 - x»^) (ct>^ + r~) 

 ^ R a -h n) e^ icj,' -{- r-y- 



Comparing this with the geodesic curvature 1/p of the line of force v' = 

 \l//(j) given by (58), we conclude that 



(91) i = i±l'. 



p K 



Hence, 



Theorem 19. Property IV'. For an " n " system, at any point on 



the surface, the geodesic curvature of the line of force is equal to {n + 1) 



