158 



LIPKA. 



of trajectories, and that any triply infinite system of curves on a 

 surface possessing these five properties may be considered as generated 

 by the motion of a particle in a unique field of force. 



We next point out how an additional property serves to characterize 

 the motion when the field of force is conservative. 



Another part of this paper presents briefly an analogous study for 

 certain other classes of triply infinite systems of curves on a surface, 

 in particular, brachistochrones, catenaries and velocity curves in a 

 conservative field of force. For all such systems characteristic 

 properties differing but slightly from those for trajectories are derived. 



§ 2. Differential Equation of the Trajectories. 



If we choose an isothermal net of curves as parameter curves on 

 the surface 



(1) X = x(u, ■?)), y = y{u, v), z = z(u, v), 

 the element of arc length may be written 



(2) ds'~ = fi{u, v) [du- + dv'']. 



The motion of a particle on the surface may be most simply expressed 

 by the Lagrangian equations ^ 



(3) 



dt \ du / du ' dt\dv / dv 



where T is the kinetic energy 



(4) 2T = Mir^+i'), 



and cp and \{/ are the components of the force given as functions of the 

 coordinates u, v.^ Introducing the value of T in (3), we get the 

 explicit equations of motion 



(5) 



(t> — - (l^uU- + 2lJ.yUV — fJLuV^) 



2 See E. T. Wliittaker, Analytical Dynamics, p. 39. 



3 Throughout the paper, dots refer to derivatives with respect to t (time), 

 primes refer to total derivatives with respect to u, and literal subscripts to 

 partial derivatives. 



