J 04 



/{u, t f t) ihi 3. (1 f ^r) du I f{n\ t) q> {.v) dx . . (17) 



If HOW we work (his out uud retain the terms up to llie fust ordei' 

 in T and it' we take (15) into consideration and if after division 

 by T, we make t approach zero, we obtain 



To the function of frequency for the velocity the extended equation 

 of diffusion is thus applicable, where 'V plays the part of coefficient 

 of diffusion. The equation is quite of the same form as that for the 

 Brownian motion under the influence of a quasi-elaslic force ( — u 

 or — s) (cf. also § 4). if we apply (18) to determine tlie stationary 

 condition we have 



d .'> dY 



= /?.-(«ƒ) + -t4 

 Ou 2 Om 



from which follows 



/3 'A /3 



;» di 







This last term becomes infinite for xl = oo, consequently the inte- 

 gration constant must be taken c, = 0. 



For the law of distribution we thus find the Maxwki,l division 

 of velocity quite independent of the initial condition. Moreover 

 Rayleigh has carefully investigated this question for his particular 

 example. He has deduced a similar equation for a particle in a 



highly rarefied gas, where only the constants jiand - have another 



meaning (cf. loc. cit.). It goes without saying that if one starts 

 from the equation of v. d. Waai.s-Snethlage, one arrives at the 

 conclusion that the division after long periods is noHhat of Maxwell, 

 and that there does not even exist a stationary division of velocity. 

 And on this point also these investigators thus come into conflict 

 with the statistical mechanics of Gibbs, which is the starting-point 

 of their reasonings. 



It may further be observed that for a particle beginning with a 

 velocity zero, as long as ii is still small with respect to the velocity 

 of the particles, which collide against it, we get as Rayleigh has 

 demonstrated 



