41] SURFACES OF EQUAL ACTION. 139 



particle projected normally to one of these planes with the energy 

 (kinetic) h is a straight line normal to these planes, and the velocity 

 is constant. 



In order to find solutions suited to surfaces of equal action 

 having other forms, we should require to find other particular 

 solutions of equation 114), which would take us too far into the 

 subject of partial differential equations. Whatever the nature of the 

 surfaces, since the velocity in all the motions considered is constant, 

 the action is proportional to the distance traversed and consequently 

 if we measure off on the normals to a surface of constant action 

 equal distances, the locus of the points thus obtained will be another 

 surface of equal action, or all the surfaces of equal action are so- 

 called parallel surfaces. 



Next, suppose we have a single particle of mass unity under 

 the action of gravity. Then 



w= ge , 



and our equation is 



TT l[/dA\*. /dA\* , /d-4\ 2 ) 7 



120) H = ~ + + + g, = ft. 



We may find a solution 



A = ax + by + <p(0), 

 where 



a 2 + 6 2 + IV 2 (*)] + 2 (ge - ft) = 0, 

 or 



Making use of this value we have 



121) A = ax + ly + y2(h-gz)-(a 2 +b 2 ) dz. 



Equations 107) become 



, dA . dA -, 



' 



giving the velocities in terms of the position of the point. These 

 are first integrals of the equations of motion. Equations 108) become 



123) 



