The Equip artition Law hi a System of Particles. 585 



density of distribution will evidently remain uniform if the 

 number of points entering any such cube per second is 

 equal to the number leaving. Consider now the two parallel 

 bounding surfaces of the cube which are perpendicular to 

 the fa axis, one cutting the axis at the point fa and the other 

 at the point <^> l + dfa. The area of each of these surfaces is 



d fad fa . . . d-\jr l d\lr 2 d'^r B . . ., and hence if fa is the component 

 of velocity which the points have parallel to the fa axis, 



and ^^ is the rate at which this component is changing as 



we move along the axis, we may obviously write the following 

 expression for the difference between the number of points 

 leaving and entering per second through these two parallel 

 surfaces : 



^[(sli) d * 1 1 d ^** ' ' ' d f^^* ' ' ' 



Finally, considering all the pairs of parallel bounding 

 surfaces, we find for the total decrease per second in the 

 contents of the element, 



p/'dfa + B<£ 2 + "dfa 4 9yj + c^2 + "djfa + \ (i y 

 \bfa ~dfa ~dfa ' 3^1 d^L- d^s / 



But the motions of the points are necessarily governed 

 by the Hamiltonian equations (1) given above, and these 

 obviously lead to the relations 



&c. 



So that our expression lor the change per second in the 

 number of points in the cube becomes equal to zero, the 

 necessary requirement for preserving uniform density. 



This maintenance of a uniform distribution means that 

 there is no tendency for the points to crowd into any particular 

 region of the generalized space, and hence if we start some 

 one system going and plot its state in our generalized space, 

 we may assume that, after an indefinite lapse of time, the 



