SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 173 



with respect to the arcs of the isothermal parameter curves as we move 

 out on the surface from 0. Furthermore, although the parameter 

 curves seem to enter this relation, (71) is really an intrinsic property of 

 our system, for it is evidently true for any and every set of orthogonal 

 isothermal curves that may be chosen. ^Ye may now state 



Property V. Construct any isothermal net on the surface. At 

 any point this net determines two orthogonal directions in which 

 there pass two isothermal curves of the net and two hj^erosculating 

 curves of Property III. If pi, po, Ri, R-i are the radii of geodesic 

 curvature of these four curves, Si, S2, the arc lengths along the iso- 

 thermal curves, and co, the tangent of the angle between the fixed 

 direction of Property III and the isothermal curve with arc 52, then, 

 as we move along the surface from 0, these quantities vary so as to 

 satisfy the relation 



1 ^ 1 a-(logco) _Q 



PlKi PiKo dSi dS'i 



where 



Property V thus completes the characterization. We may now 

 state 



Theorem 13. In order that a triply infinite system of curves (coi 

 iti each direction through each point) on a surface may be identified with a 

 system of dynamical trajectories under any positional field of force, 

 the given system must possess Properties I, II, III, IV, V. 



§ 9. Special Case. — Conservative Forces. 



If the field of force is conservative, there exists a work function 

 (negative potential) W of which the force components 0, xp are the 

 derivatives: hence 



(72) = Wu, ^ = n\; or i/'„ = 0„. 



We may interpret this relation geometrically by noting that if the 

 conic (45) is to be a rectangular hyperbola, the sum of the coefficients 

 of ^- and 'ff- must be zero; hence ^^ = <^„; and conversely. Therefore, 

 we have 



Theorem 14. If the field of force is conservative, the locus of the 



