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



under the action of the given forces with the constant energy h is 

 the vector differential parameter of the action A, and therefore, by 

 the properties of lamellar vectors ( 31), the velocity of a particle 

 moving in this manner is normal to all the surfaces of constant 

 action, and is inversely proportional to the distance between two 

 infinitely near surfaces of constant action. Otherwise expressed, if 

 from all points of any surface particles be projected normally with 

 the same energy h, their paths will always be normal to a set of 

 surfaces, and the action from one surface to another will be the 

 same for all the particles. This theorem is due to Thomson and Tait. 1 ) 

 Suppose first there are no forces acting, then equation 112) 

 becomes 



which is satisfied by the linear function 



115) A = ax + ~by + C8 9 

 if 



116) a 2 -f 6 2 + c 2 = 2wft. 



In virtue of this last equation only three of the constants 

 a, 6, c, li are arbitrary. Suppose we take a, 6, ft, then we have 



117) A = ax + ~by- 

 Then equations 107) or 113) are 



118) mx' = a, my' = 1, mz' =-\/2mJi - (a 2 



which are first integrals of the equations of motion, showing that 

 the motion of the point is uniform. Equations 108) and 109) are 



dA as = 



~ 



dA lz 



119) ~n=y- 



y%mh-(a*-\- & 2 ) 

 3 J. m# 



/ / 



6 v/i 



^^ l/^-mTj r 2 -l-?i^ 



The first two of these equations are the equations of the path, 

 showing it to be a straight line, while the last gives the time. By 

 means of it we may find 2 as a function of the time, and from the 

 first two x and y. Thus 119) are the integral equations of the 

 motion. 



Corresponding to this solution, a and & being constants, the 

 surfaces of constant action are parallel planes. The path of any 



1) Natural Philosophy, 332. 



