( 633 ) 



random for each of tlie n molecules of a certain quantity of gas ; 

 the "point of velocity" of the molecule was every time placed in 

 the volume element extracted. The n velocities which have been 

 extracted chanced to be such that the sum of the energies of the 

 molecules has a definite value L. What distribution of the velocities 

 among those n molecules is now a posteriori the most probable? 



The most probable is therefore the distribution of Maxwell. But 

 this is a problem without importance for the gas theory. For it 'is 

 easy to see that the mean velocity indicated by the slips of paper 

 in the box is intinitely large, so that if from this n slips of paper 

 are taken at random, in general the mean velocity, which is indicated 

 by them, is also infinitely large, and there is only an infinitely small 

 chance, that the energy of the n molecules becomes finite. If we 

 now see that of every finite gas-mass the energy is finite, we cannot 

 assume that the velocities would have been assigned to the molecules 

 in the way mentioned above. The chances a priori for every velocity 

 must, therefore, not be considered as equal. 



The mean velocity in the box may be calculated as follows. 



If in the unity of volume there are c points of velocity, then in 



a spherical shell with radius r and thickness dr there are : 4 Jt r' c dr. 



The sum of the velocities now is 4 rr r' c dr, and so for a sphere 



with radius r = jt r* c. The number of points of velocity in the 



4 3 



sphere is - jt r' c, so the mean \ elocity = — r. 



o '4: 



For the whole space, therefore, the mean velocity is infinitely 

 large. In this way it is proved that the hypothesis of the equality 

 of the chances a priori is inaccurate, and so also the result that 

 Maxwell's distribution of velocities is the most probable state. 



^ 2. Of course nothing is said here in derogation of the proof, 

 tliat Maxwell's law holds in the stationary state, which Boltzmann 

 gives in the § ^ preceding § 6 and in ^ 7. But it is incorrect to 

 speak of transition of probable to improbable states when the meaning 

 is from stationary to non-stationary states. This incorrect view gives 

 rise to wrong considerations when Boltzmann discusses the fiction 

 of the reversal of the molecular velocities in the last part of ^ 6. 



It is assumed there, that a gas has originally a "molecular-unge- 

 ordnet" but "improbable" distribution e.g. all molecules have the same 

 velocity. The gas moves now to the stationary state Avith Maxmell's 

 distribution of velocities. But before it is reached, all velocities are 

 reversed, which causes the same conditions to be passed through 

 but now in reversed succession. This will cause H to increase. Is 



