176 LIPKA. 



where G is defined as in (9), or 



£(„- + ,) = ("---"'f " + '■"'. 



m 



Differentiating this last expression, we get 



(80) - (Tr„ + ir,r') = 



g4- \ (IT\. - TTV) (1 + v") ] - 0' (fr, - Wy) (l+r'2) 

 dii ( ) 



If we introduce the components of the force 



(81) ct> = Tf „ lA = W,, 



and solve for 6", at the same time replacing 2/m by n, we find 



(82) (V' - 0/) G'= G j {^Pu + n\,ct>) + (^/'r - </). + nKiy - 7i\,4>y 



v" 



0+f/ „ , 



for the required differential equation, where 



71 = 2 corresponds to dynamical trajectories 

 n — — 2 " " brachistochrones 



w = 1 " " catenaries, 



and G and 6" are defined by (9). ^Ye shall designate the curves 

 defined by (82) as an " ?t " system. AYe 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. 



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

 termed " velocitv " curves. Thev are defined dvnamicallv as follows: 



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

 starting with 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, Ave note that the 

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

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

 velocity from equation (8). Using the relation 



6-2 = c''-Hu--\- i^) = t-\y' (1 + V'~), 



