312 Prof. L. Boltzmann on the Assumptions necessary 



also those of the molecules of the second order ; so that 

 the conditions at once become completely altered. 



If we assume Maxwell's law of distribution of velocities, by 

 putting /(v) =Ae~ az \ F(V) = Be~ 6V2 we find, by precisely the 

 same calculation as that of Prof. Tait, that there is no exchange 

 of energy between the molecules of the first kind and those of 

 the second, if the mean energy of both is the same — which, 

 (following Prof. Tait) we will call Maxwell's Theorem. 



So far all Prof. Tait's conclusions are, without doubt, cor- 

 rect. But it does not yet follow from this that the existence 

 of Maxwell's law of distribution must be assumed in order to 

 prove Maxwell's theorem. It may rather be shown that, 

 whatever the masses and ratio of diameters may be, if only the 

 molecules of the first order come generally into collision with 

 those of the second order, then Maxwell's distribution of velo- 

 cities is spontaneously brought about both amongst the former 

 and also amongst the latter molecules. 



In this it is not even necessary to assume that the molecules 

 of the first order are generally in collision amongst themselves, 

 nor that the molecules of the second order are in collision 

 amongst themselves. The only assumptions are : that both 

 the molecules of the first and also those of the second order 

 are uniformly distributed over the whole space; that through- 

 out they behave in the same way in all directions; and that the 

 duration of the impact is short in comparison with the time 

 between two impacts. I have given* the proof of this in my 

 paper on the Thermal Equilibrium of Gases acted on by Ex- 

 ternal Forces, at the conclusion of § 1 ; but as I have there 

 only briefly indicated the mode of calculation and have only 

 given the result, Prof. Tait has probably quite overlooked the 

 passage, and I will be more explicit on the present occasion. 



Let the molecules of the first and second kinds of gas be at 

 the beginning of the time (t=0) uniformly distributed in a 

 space enclosed by rigid perfectly elastic walls ; but let the 

 distribution of energy among them be altogether arbitrary. 

 Exactly as before, let 



4cirv 2 f(v } o) dv 



be the number of molecules of the first kind of gas in the unit 

 volume whose velocities lie between the limits 



v and v + dv (6) 



In exactly the same way, let 47rV 2 F (V , o) dV be the number 

 of molecules whose velocities lie between 



VandV + dV (7) 



* Wien. Sitzb. vol. lxxii., October 1875. 



