Partition of Kinetic Energy. 103 



most general expression for the law of distribution is 



f(.v, y, u, v) dx dy du dv, (9) 



signifying that the number of particles to be found at time t 

 within a prescribed range of phase is to be obtained by inte- 

 grating (9) over the range in question. But such a distri- 

 bution would in general be unsteady. If it obtained at time t, 

 it would be departed from at time t f y and vice versa, owing to 

 the natural motions of the particles. The question before 

 us is to ascertain what distributions are steady, i. e. are main- 

 tained unaltered notwithstanding the motions. 



It will be seen that it is the spontaneous passage of a 

 particle from one phase to another that limits the generality 

 of the function /. If there be no possibility of passage, sav, 

 from the phase (x f , y', u', v f ) to the phase (x, y, u, v), or, as 

 it may be expressed, if these phases do not lie upon the same 

 path, then there is no relation imposed upon the corresponding 

 values of/. An example, given by Prof. Bryan (7. c. § 17), 

 well illustrates this point. Suppose that V = 0, so that every 

 particle pursues a straight course with uniform velocity. 

 The phases (a/, y', u\ v f ) and (x, y, u, v) can lie upon the 

 same path only if u'=u, v' = v. Accordingly/ remains arbi- 

 trary so far as regards u and v. For instance, a distribution 



/ (u, v) dx dy du dv (10) 



is permanent whatever may be the form of/ understood to 

 be independent of x and y. In this case the distribution is 

 uniform in space, but uniformity is not indispensable. Suppose, 

 for example, that all the particles move parallel to x, so that 

 /vanishes unless i' = 0. The general form (9) now reduces to 



f K x, y, u) dx dy du ; (11) 



and permanency requires that the distribution be uniform 

 along any line for which y is constant. Accordingly,/ must 

 be independent of x, so that permanent distributions are of 

 the form 



/(y, u) dx dy du, (12) 



in which / is an arbitrary function of y and u. H either 

 y or u be varied, we are dealing with a different path (in the 

 sense here involved), and there is no connexion between the 

 corresponding values of / But if while y and u remain 

 constant, x be varied, the value of/ must remain unchanged, 

 for the different values of x relate to the same path. 



Before taking up the general question in two dimensions, 

 it may be well to consider the relatively simple case of motion 

 in one dimension, which, however, is not so simple but that 





