SUKFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 177 



we may write equation (S) in the form 



(80 v"= \ [(^ + X„) - (0 + X„).'] [1 + v'^. 



Equation (8') holds for any trajectory and along this the velocity s 

 varies from point to point. Now, if in (8') we replace 1/ s^ by a 

 constant c, we get 



(8-) /'= c [(^ + X„) -(</, + X„)/] [1 + v' % 



a differential equation of the second order representing a system of oo ^ 

 curves, one through each point in each direction. Each of these 

 curves, therefore, has the dynamical property used above in defining a 

 velocity curve. ^ 



For each constant value assigned to the speed s, we get a velocity 

 system, and the totality of oo^ systems obtained by var^'ing s consti- 

 tute a complete velocity system of oo ^ curves on the surface. To find 

 the differential equation of the complete velocity system, we, therefore, 

 differentiate (8") and eliminate the parameter c. Writing (8") in the 

 form 



c G 



we get by direct differentiation, 



4 



for the required differential equation of a complete velocity system. 

 We now note that this is the form taken by equation (82) if n = 0. 

 Hence, we may state that an " ?i " system represents a velocity system 

 when n = 0. For a velocity system in a conservative field of force we 

 merely add conditions (81). 



§11. Geometric Characterization of "n" Systems. 



Let us now find the geometric properties of the system defined by 

 (82). Since this equation has the form of equation 



8 A velocity system of cxjs curves is characterized geometrically by the fact 

 that the locus of the centers of geodesic curvature of the oo i curves which pass 

 through a given point, is a straight line. See the author's paper "Geometric 

 characterization of isogonal trajectories on a surface," Annals of Math., 2d series, 

 15 (1913), pp. 71-77. For further discussion of velocity systems see the 

 author's paper "Note on velociti/ si/stems in curved spece of n-dimensions," Bull. 

 Am. Math. Soc. 2d series, 27 (1920), pp. 71-77. 



(82') {■4^-4>v')G'=G 



•^„ + (;//„-<^„)/-(^y2 



+ 



2^+^-3^ 



1+7/2 



