41] 



PARTICLES UNDER GRAVITY. 



141 



we obtain the action as a function 

 whichever path is described , 



of the coordinates x and , 



188) ^ 



Putting A equal to 

 a constant we obtain 

 the curves of equal 

 action which cut 

 the parabolic paths 

 orthogonally. This 

 problem is treated 

 in Tait's Dynamics, 

 219. 



This problem is 

 geometrically equi- 

 valent to that of 



streams of water issuing from holes in a vertical side of a tank, for 

 it will be proved in Chapter XI that the velocity of water so issuing 

 varies with the height in precisely the manner above prescribed, the 

 parabolic paths corresponding to the jets of water. It is easy to 

 show that all these parabolas touch a common line making an angle 

 of forty -five degrees with the vertical, and that the curves of equal 

 action have cusps on this line. (Fig. 28.) 



As a further example, let us add an arbitrary constant to the 

 value of A (which may always be done) writing, 



Fig. 28. 



= ax - 



129) 



130) x + - 

 y 



If now x = 0, = 2 Q , A vanishes, thus one of the curves of 

 equal action shrinks to a point. The problem is then that of 

 particles projected in a vertical plane from the same point (0, %) 

 with the same velocity. The equations of the various paths and the 

 action corresponding contain the parameter a. Eliminating a we 

 obtain the action in terms of the position, 



131) 4-S 



The various paths here treated have a parabolic envelope as described 

 in 35. The curves of equal action here again have cusps on the 



