104 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



outside the radical is greater than that under it, both values of t 2 

 will be positive, even if the lower sign is used, therefore there will 

 be two real possible positive values of t. 



To determine , the angle of elevation, we have 



Vs = ^n Sm a > 



tan a = ~ 



Vx 



and inserting the two values of t we get two possible elevations. 

 Thus we find that the aim is completely determined (though not 

 uniquely in this case) by the terminal positions and the velocity of 

 projection. 



For the action we obtain 



15) 



A = 



= m 



- *)} dt 



Using the values of V z and t found above we obtain two values of 

 the action different for the two paths. Thus there are two possible 

 natural paths, differing from each other by finite distances, for only 

 one of which is the action least. Both however have the property 

 that between two points sufficiently near together the action is less 

 than for any infinitely near path. 



In case the radical in 14) vanishes, that is 



the two roots t 2 are 

 equal and there is 

 only one course. 

 The terminal point 

 x lf #! then lies on 

 a parabola whose 

 vertex is vertically 

 above the point of 

 projection (Fig. 25). 

 It is easy to see 



Fig 25 that this parabola is 



the envelope of all 



possible paths in this vertical plane starting from the same initial 

 point x Q1 with the same velocity t' . For it is the locus of the 



