Illustration of the Theory of Gases. 441 



When t = 0, 



%=—n~ l rsm$, dxjdt — u=r cos 6 ; 



and we may regard the amplitude and phase of the vibrator 

 as determined by u, rj, where 



u = r cos 6, rj—r sin 6. 



Any distribution of amplitudes and phases may thus be ex- 

 pressed by f(u, rj) du drj. 



If we consider the effect of the collisions which may occur 

 at £=0, we see that u is altered according to the laws already 

 laid down, while t] remains unchanged. The condition that the 

 distribution remains undisturbed by the collisions is, as before, 

 that, for every constant n,f(u, tj) should be of the form e~ hu2 , 

 or, as we may write it, 



/K'))=xW«- s «'. 



But this condition is not sufficient to secure a stationary state, 

 because, even in the absence of collisions, a variation would 

 occur, unless /(w, rj) were a function of r, independent of 6. 

 Both conditions are satisfied, if x{v) — ^ e ~ hri% ^ where A is a 

 constant ; so that 



f(u, rj) du dn = A e ->K" 2 +» 2 ) du drj = 2ttA e~ hr * r dr. 



. Under this law of distribution there is no change either from 

 the progress of the vibrations themselves, or as the result of 

 collisions. 



The principle that the distribution of velocities in the 

 stationary state is the same as if the masses were free is of 

 great importance, and leads to results that may at first appear 



. strange. Thus the mean kinetic energies of the masses 

 is the same in the two cases, although in the one case there is 

 an accompaniment of potential energy, while in the other 

 there is none. But it is to be observed that nothing is here 

 said as to the rate of progress towards the stationary condition 

 when, for instance, the masses start from rest ; and the fact 

 that the ultimate distribution of velocities should be inde- 

 pendent of the potential energy is perhaps no more difficult 

 to admit than its independence of the number of projectiles 

 which strike in a given time. One difference may, however, 

 be alluded to in passing. In the case of the vibrators it is 

 necessary to suppose that the collisions are instantaneous ; 

 while the result for the free masses is independent of such a 

 limitation. 



The simplicity of/ in the stationary state has its origin in 

 the independence of 6. It is not difficult to prove that this 



