( 396 ) 



presents the dejisity in each of llie i)üi]its of the original area 



which now come to the same thing. 



Now this P' becomes, as we saNv, in conrse of time a constant 



and so >S); minimnm. This qnantity might be called entropy of ])hice; 



"entropie grossière" seems less appro[)riate, because the elemejits of 



extension are infinitely small. This final approach to a minimum 



must not be taken as a continual decrease. Jt is easy to see that the 



oi-iginal function of density might be chosen in such a way that 



there are also times, at least in the beginning, at which the densities 



P' rather diverge than draw nearer to each other. Then the ([uantity 



dSn dH 



S.J would increase; so — - is not like Boltzmann s — negative all 

 ' Jt dt 



through. 



If, however, we compare times, in which (list an angle 2jr, then 

 4rr etc. is occupied, ^ve may say with a pretty high degree of 

 certainty, that >Sp has always diminished. If now instead of a hori- 

 zontal area an arbitrarily chosen ensemble is considered, the above 

 reasoning may be aj)|)lied foi' e\ery horizontal elementary area from it. 

 So now too the ensemble is tinally uniformly spread over the ecliptic, 



and llie (Hiantitv >% = I i^'%i^'<W becomes minimum. Now, however. 







P'(//r= ^ j7\//(/(o or r' = 2:: Ipdo), QVL^vy time integrated over 



all w's which fall uithin the iKH'izontal area determined by d/. 

 This P' becomes tinally constant. The motion of the ensemble in 

 this more general case is expressed by tig. 2. 



The above mentioned inaccuracy in Poincakk's reasoning is this: 



he considers the quantity 6'=: I I T Un/ Pdldoj, in which he integrates 



Avith regard to / from to 2.t. So the P from this formula has 

 evidently arisen by summation for the different values of / whicli 

 come to the same thing, but iiot by integrating with respect to to. 

 Hence this /"' is the sum of the densities of the elements obtained 

 by taking a detinite to, and then successively /, / -f- 2.t etc. In tig. 2 

 these elements are cross-hatched for one value of oj. Howe\'er, 

 'PoiNCARÉ assumes further that tinally for t =: cc this P, or i-ather 

 this ^P no longer depends on /, but only on to^). This seems 

 to me incorrect. For then for every vertical elementary area in 

 tig. 2 the sum of the densities in the elements, which are every 



-) Reflexions sai hi Theorie ciuélique des gaz; p. 381, p. 885 etc. 



