( 392 ) 



Avliolc region occupied hy the eiisemhlf) is coiistiuit if tlieexlenia! circiiiii- 



ÖA'/ 

 stances are nnciiangecl and tiie relation J^ — — exists (tor w liicli 



'li-i \ 

 A/ = . In this PoiN'CARK takes as variables the coordinates 



dt J 



and velocity-components of the molecules; so the f(uantity referi-ed 

 to above dillers from the (piantilv n introduced by Gihbs only by 

 a constant factor. 



(iiBps shows that in a canonical ensemble — ?; has l he properties 

 of the entropy, Poincakk calls the quantity S itself enti'Opy, also for 

 an arbitrary ensemble. Hence this quantity will have to decrease, 

 where — ^i increases. 



The property in question may i)e derived as follows: P has the 

 properties of a density, so: 



dF ^JJ'X; ^ ÖP/ dX; 



- ^ 2^ X; ^ as 2 ^-- = 



d^ d.Vi d.v; \ d.v; 



Now, an arbifi'ary funciion of /* will also i)ehave as a density. 

 Namely : 



ö/"(^) dP „ ,^^ ^.. ÖP , dF dXi 



= _ V ,,.•■ ,7.) f - ^ f in '"'• =-^ '^^^ . 



O.i-i d.Vi O.vi 



So the /(/'*) also satislies an equation of the same foi-m as the 7^ 

 itself, which equation represents the extension of the wcllkno^\ll 

 equation from hydrodynamics: 



do doit dov doir 

 dt d.v dy ^ dz 



to a sj)ace of i} dimensions. 



Now /S'= ij'lo(l F(/t, (in which (It — d.v., . . . (I.r,,^) — I /(P) dr, 



is the integral of such an /{P), integrated over the whole extension 



occiqtied by the ensemble. To ascertain the change of ^' with 



the mol ion of the ensend)le, we must every time integrate over the 



vai'iable space (Ihotigh constant in size), over which the j)hases 



extend or whei-e /* and also ƒ(/'') have values. So we can jterfeclly 



(75 

 compai'e with the ijicrease in lime ot a (pianlity ot li(piid. 



dt 



taken over all the places where it is. This increase, howevei-, is 



equal to zero. 



