SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE 



175 





muiimiun, 



and the paths are catenaries. 



For a given constant of energy, h, (74), (75), or (76) will give oo2 

 curves, one through each point in each direction on the surface.'' 

 If we allow h to vary, we shall get triply infinite systems of curves: 

 complete systems of dynamical trajectories, brachistochrones, or 

 catenaries. The systems defined by (74), (75), and (76) may be con- 

 sidered as special cases of the system defined by 



(77) 



/"(Pi) 



JiPo) 



(W -\- h)~ ds = minimum, 



where we have trajectories, brachistochrones, or catenaries, according 

 as m =1,-1, or 2. 



Replacing ds by e Vl + i'' " du, and applying the Euler condition 

 for the vanishing of the first variation, to 



(78) 

 viz., 



/<"• 



m n 



+ h) 2 e^ VI -\- v'"^ du = \ H du = minimum, 



H^IU — V' Hv'v — V" Hytvl = 0, 



// 



we 



find 



(79) V 



-1 



/oj/(ir+/02+x 



%(ir+/02+x 



1+.' 



as the differential equation of the system of oo- curves. To find the 

 differential equation of the system of oo^ curves, we must differentiate 

 (79) and eliminate h. This is most readily done by writing (79) in the 

 form 



m TI\ - WJ _ v"- (X. - X„/) (1 + in _ G 



9 



W + h 



1 + /2 



1 + v' 



7 These systems of co 2 curves form special cases of the extremals connected 

 with a variation problem of the form /Fds = minimum, where F is a function 

 of the coordinates. Such systems, termed "Natural S^jstems," hiivc been 

 characterized geometrically by the author in "Natural famihes of ^^^'fy/J-f 

 general curved space of N dimensions;' Trans. Am. Math. Soc, vol 13 {l.ni), 

 pp. 77-95. The author has also characterized these curves in a diHerent way 

 in "Some geometric investigations on the general -problem of dt/namics Fro- 

 ceedings of the Am. Academy of Arts and Sciences, Vol. 55 (1920), pp. 285-d/J. 



