SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 169 



^ '^^ ^ 3(1 + CO!/) 



We may finally state 



Theorem 9. The most general triply infimte system of curves on a 

 ■surface possessing Properties J, II, and III is defined by a differential 

 equation of the form 



(56) (co- v')G'= 6'(7o+ 7i^''+ 72*'"- 3i/') 



involving four arbitrary functions 70, 71, 72, w of u, v. 



By comparison of (56) with the differential equation of the tra- 

 jectories 



(11) (^ - 0/)G"= G'{(iA»+ 2X,.c/)) + (^„- 4>u+ 2\,rp- 2X„0)/ - 



(cA, + 2X„^)/2-30r"}, 



involving only two arbitrary functions of u, r, we note that Properties 

 I, II, and III are not sufficient to characterize the system of tra- 

 jectories. We may here note the similarity in form of equations (11) 

 and (56). 



§ 7. The Lines of Force. Curves with Property I, II, III, IV. 



On the surface, a line of force is a curve such that its tangent line 

 at any point has the direction of the force vector through that point. 

 The lines of force thus form a simple system of 00 1 curves defined by 

 the differential equation 



(57) v'= xp/cl>. 



Employing (39), we find for the geodesic curvature of the line of force 

 passing through the point 0, 



(r^. 1 ^ 4>'^u- r-4>v'\- 4>M4'v- <t>u) - (X.0 - X„;^) (0^^+ 4^') 



How does this compare with the curvature of the unique hyperoscu- 

 lating trajectory passing through in the direction of the force vector? 

 The value of r" corresponding to a hyperosculating trajectory is given 

 by (43); introducing this and the direction v'= xf/fcj) into (39), we find 

 for the required geodesic curvature, 



/^Q^ i - '^''^«- '^'^'■+ ^'A('/'>.- <Pu) - (K4> - XuV') (</>'+ f-) 

 ^^""^ R 37(^M^W 



