106 



Lord Ravleio-h on the Law of 



In the determinants of (19), (20) the motion is regarded as 

 a function of x, y, x\ y', and the three quantities which do 

 not appear in the denominator of any differential coefficient 

 are to be considered constant. This was also the understand- 

 ing in equations (8), from which we infer that the two deter- 

 minants are equal, being each equivalent to 



_^S rf«S 



dx dx 1 3 dx dy ] 



^ 2 S rf«S 



dx' dy ' dy dy' 



Hence we may write 



(21) 



dx dy du dv= dx' dy' du dv', 



■ ■ (22) 



an equation analogous to (15). By the same reasoning as 

 was employed for motion in one dimension it follows that, 

 if the distribution is to be steady, f(x, y, u, v) in (9) must 

 remain constant for all phases which lie upon the same path. 

 A distribution represented by 



f(E)dxdydudv, (23) 



where 



E = ^ 2 + ir 2 + V, 



(24) 



will satisfy the conditions of steadiness whatever be the form 

 of /; but this form is only necessary under the restriction 

 known as Maxwell's assumption or postulate, viz. that all 

 phases of equal energy lie upon the same path. 



It is easy to give examples in which Maxwell's assumption 

 is violated, and in which accordingly steady distributions are 

 not limited to (23). Thus, if no force act parallel to y, so 

 that V reduces to a function of x only, the component velocity 

 v remains constant for each particle, and no phases for which v 

 differs lie upon the same path. A distribution 



/(E, v) dxdydudv (25) 



is then steady, whatever function /may be of E and v. 



That under the distribution (23) the kinetic energy is 

 equally divided between the component velocities u and v is 

 evident from symmetry. It is to be observed that the law 

 of equal partition applies not merely upon the whole, but for 

 every element of area dxdy, and for every value of the total 

 energy, and at every moment of time. AVhen x and y are 

 prescribed as well as E, the value of the resultant velocity 

 itself is determined by (24) . 



