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ƒ_. ag — ö(£ a -H-, 2 -R 2 \ where a and ft are real constants, so that tor this 

 system Maxwell's distribution of velocities holds for any small part 

 of the vessel, and also the density is constant throughout the vessel 

 for one definite velocity. That tor the great majority of the systems 

 the distribution of place and velocity (litters little from the maximum 

 occurring one may be shown in the same way as it is shown when 

 the distribution of place and velocity i> considered separately. 



It is, however, easy to see that if an ensemble arises, not a micro- 

 canonical one, indeed, but one, for which the mean density becomes 

 constant for finite but small extension elements, by approximation 

 the same resnlt will be obtained as we have for a microcanonical 

 ensemble, viz. uniform distribution throughout the vessel and Maxwell's 

 distribution of velocities, and that with greater accuracy as the 

 elements are smaller. So we shall have a kinetic derivation of the 

 2 nd law of thermodynamics, if we can show that an arbitrary en- 

 semble of systems with a detinite kinetic energy passes into such a 

 "rough" microcanonical ensemble. So this has again led us to the 

 quantity called "entropie grossière" by Poincaré", for if 77, the mean 

 density over the elements rf, becomes constant, 2 fllog TI <f or the 

 entropie grossière decreases. It seems to me that we might demonstrate 

 in the following way that // becomes constant in course of time. 



Let us in the first place once more consider the ensemble of 

 planets or one-dimensional moving molecules discn>sed in $ 2 and § 3 

 of the above-mentioned paper. It was shown that this ensemble 

 moves in such a way that finally all places occur equally frequently. 

 This was the case for all the velocities together and happened just 

 because all kinds of velocities occurred for the systems. If, however, 

 the total amount, over which the velocities of the systems (planets 

 or molecules) extend, is divided into small, but finite portions, it 

 will also hold for these amounts separately, if we only take the 

 time long enough. So when these amounts extend from <u, to 

 co x -r Aw,, from w, -f Aeo to cj, -f 2Aio etc. the systems with velo- 

 cities lying between e>, and co, 1- Ac> will finally be uniformly 

 distributed over all the values / lying between and 2.t; in the 

 same way the systems with velocities between w, 4-Aai and «vf-2 Ato 

 etc. Each of the horizontal strips of fig. 1 lying above each other 

 contains then the same number of representing points. 



If instead of an ensemble of single planets or single molecules we 

 take an ensemble of systems of n molecules each but disregard the 

 collisions, the same reasoning wil! hold. The whole of the representing 

 points now moves, however, in a ('^-dimensional space, and instead 

 of the axis of distances and the axis of velocities we get now the space 



