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



These are the equations of the path completely integrated, 

 showing that it is a parabola in a vertical plane. The equation 109) is 



124) = 



giving the time. 



From the last equation we may obtain z in terms of t, and, 

 from the two preceding, x and y. Thus the problem is completely 

 solved, the constants, a, &, and k, being determined by the terminal 

 conditions. 



Suppose we put & = 0, then 121) and 123) give 



A 

 125) A = ax - {2 Qi-ge) - a 2 } 2 , 



126) 

 or 



If we consider motions for which a is a constant, but a 1 has 

 different values for the different motions, all the parabolas are 

 obtained from a single one by displacing it horizontally. The curves 

 of constant action, 



127) {3# (ax - A)} 2 = {2 (h - ge) - a 2 } 3 , 



are semicubical para- 

 bolas, similarly ob- 

 tained by displacing 

 a single one hori- 

 zontally, and cut the 

 parabolas at right 

 angles (Fig. 27). 



The same solu- 

 tion of the differ- 

 ential equation may be adapted to the treatment of other problems. 

 If we put #j = 0, x and A will vanish simultaneously, or one of the 

 curves of equal action will be the vertical line x = 0. Thus we 

 have a solution of the following problem. Particles are projected 

 horizontally in a vertical plane from points on the same vertical line 

 with such velocities that the total energy is equal to h, the same 

 for all. The different parabolic paths contain the parameter a which 

 changes from one to another. The action along any path contains 

 the same parameter. Eliminating a between equations 125) and 126) 



