( 393 ) 



H(»\ve\(M\ wlieii N= | I* /e;/ /' i/.i\ ... (/■>'„ is c'oiishiiii, the CMisemble 



caniiol move in (he direelioii lo the inierocaiiüiiieal distribiUioii. 

 For (hen /* wonUl become constant all o\ci' the j)hase-exteiisioa 

 in course of time, and so the integral would get a minimum value. 



Now, however, (he ((uesdoa suggests itself: when the fimetion ;S' 

 or 1] is constant, is (here not another quantity charactei'ising the 

 ensemble, which bv its \arialion in a certain direction indicates the 

 nu)tion of the ensemble in the direction of the nnifoi-m distribution 

 of |)lace and Max\velt;s distribution of velocities? For this motion 

 is hardly open to doubt, atul in a special case such a function has 

 been found for one system by Boltzmann in the quantity //. 



PoiNCAHK supposes lie has found such a fuiiction in his "entropie 

 grossière", a quantity of the same form as ,S', but in which the elements 

 of the area over which the summation is made, are not taken infinitely 

 small, but so small that practically we cannot distinguish between 

 sj'Stems lying within the same element. This (juantity may, therefore, 

 be represented l>y .^nioan.d, in which J represents the element 

 and n the mean density in it. In contradistinction with this entropy 

 S is called the "entropie line", and it may easily be shown that 

 the "entropie grossière" is always smaller than the "entropie line". 

 It is less easy to sec that the "entroi)ie grossière" gradually decreases, 

 nor does Poincare j)rove this. For it is not easy to see that the 



(pundit V *S = I I F Irxi F (U (ho, of which he tries to prove in some cases 



that it has decreased, represents an "entropie grossière", while the proof 

 too rests on an assumption which is unjustitied in my opinion. It 

 is true that we shall demonstrate turther on, that there are quan- 

 tities of this form which decrease, but for them the name of "entropie 

 grossière" is not very appropriate, as the elements of the extension over 

 which the summation is made are just as well infinitely small, 

 though of lower order of magnitude than the original elements. 



§ 2. A very suitable iiitroduction in the theory of gases is supplied 

 by the pruh/rin of tin' .■oikiII plaiieU'^) repeatedly treated by Poincare. 

 There the problem is discussed what in course of time the distribution 

 along the ecliptic will become of a nund^er of small planets, which 

 at some time were placed in tlieii- orbit in such a \\ix\ that chaiice 

 has decided at least the distribution of the velocities. Poincark show^s 



') Gf. I.e. and also: "Calcul des Probabilités", Paris 1890 and "La Science 

 et l'Hypothèse" Paris 1904. 



