.176 LIPKA. 



where G is defined as in (9), or 



-{W -\- h) = . 



m Or 



Difi^erentiating this last expression, we get 



(80) - (ir„ + n V) = 



m 



6'|- 1 (ir„ - nv) (1 + v'^) I - G' ow - ir.o (i+z^) 



If we introduce the components of the force 



(81) </) = Wu, '^ = H\„ 



and solve for G', at the same time replacing 2/m by n, we find 



(82) (lA - (^/) 6"= G (}Pu + «X,0) + C^,, - (/)„ + 7iX,-.A - nKcl>y 



-((^, + »X„<A)«'2 + 



^ ^l + r'2 



1)" 



for the required differential equation, where 



n = 2 corresponds to dynamical trajectories 

 ?( = — 2 " " brachistochrones 



n = I " " catenaries, 



and G and G" are defined by (9). We shall designate the curves 

 defined by (82) as an " ?i " system. We may here note the similarity 

 of equation (82) for the " n " system and equation (11) for the dynami- 

 cal trajectories in any field of force. 



We may also study certain other types of curves on a surface, 

 termed " velocity " curves. They are defined dynamically as follows: 



A curve is a velocity curve corresponding to the speed So, if a particle 

 starting Avith that speed from any point of such a curve and in the 

 direction of the curve, describes a trajectory osculating the curve. 



To get the differential equation of such a system, we note that the 

 differential equation (11) of the trajectories for any positional field 

 of force was obtained by eliminating the variable component w of the 

 velocity from equation (8). Using the relation 



