174 LIPKA. 



centers of geodesic curvature of the oqI hyperosculating trajectories which 

 pass through any point of the surface, is a rectangular hyperbola. 



Combining Theorems 13 and 14, we state 



Theorem 15. In order thai a triply infinite system of curves on a 

 surface may he identified with a system of dynamical trajectories under a 

 co7iservative field of force, the given system must possess Properties I, II, 

 III, IV, V, and the additional property that the central locus of Property 

 III is a rectangular hyperbola. 



§ 10. Brachistochrones, Catenaries, Dynamical Trajectories 



and Velocity Curves. 



If the field of force is conservative, we may study certain tjpes of 

 to 3 curves on a surface other than dynamical trajectories. Among 

 these, two cases of special interest are the systems of hrachistochrones 

 and catenaries. To get the equations of these systems we proceed as 

 follows. 



Consider the motion of a particle in a conservative field of force 

 from one position Pq to another P], with the sum of its kinetic and 

 potential energies equal to a given constant. If T is the kinetic 

 energy, W, the work function (negative potential), v, the velocity, and 

 h, the constant of energy, we have 



(73) T -W = h, or h"" - W = h, or ^^ _ 2 (W + h). 



(i) If the motion takes place under the principle of least action, 

 i.e., so that 



(74) Action = 



J^(A) riPi) riPi) riPi) 



(Po) 2^^^=j(Po) "''^^=j(Po) ''^'=,M) ^'^Oy+h)ds = minivium, 



the paths are dynamical trajectories. 



(ii) If the motion takes place so that the time elapsed is least, i.e., 

 so that 



05) Ttmc =J^^^^ dt = j^^^^ ^ = j^^^^ VWTh) ^ '''''''''''''"' 



the paths are brachistochrones. 



{Hi) If the motion takes place along the position of equilibrium of 

 a homogeneous flexible inextensible string, then 



