SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 175 



J^^Pi) /•(ft) 



,„, V'ds = I ^ 2(ir + h)ds = minimum, 

 in) UKfo) 



and the paths are catenaries. 



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

 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 



J'^(Pi) ^ 1L 



{W + /;)2 ds = minimum, 

 C' o) 



where we have trajectories, brachistochrones, or catenaries, according 

 as m =1,-1, or 2. 



Replacing ds by e v 1 -f- v' - du, and applying the Euler condition 

 for the vanishmg of the first variation, to 



(78) j (W + /O 2 e^ Vl + v' 2 du =J II du = mi 



viz., 



i/« — Hv/u — V Hv'v ~ * Hv'v' = 0, 



log{W-\-h)2+\ 



log{]V+h)2+\ 



yjji+.-j, 



as the differential equation of the system of oo2 curves. To find the 

 differential equation of the system of co^ curves, we must dift'erentiate 

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

 form 



2 W + h ~ 1 + ^'' l-{-v'-' 



7 These systems of oo - curves form special cases of the extremals connected 

 wdth a variation problem of the form J'Fds = minimum, where F is a function 

 of the coordinates. 8uch systems, termed "Natural Systems," have been 

 characterized geometrically by the author in " Natural families of curves in a 

 general curved space of N dimensions," Trans. Am. Math. Soc, vol. 13 (1912), 

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

 in "Some geometric investigations on the general problem of dynamics," Pro- 

 ceedings of the Am. Academy of Arts and Sciences, Vol. 55 (1920), pp. 285-322. 



