( 391 ) 



(luaiitily // =r 1 /.//*. J(o ('onslaiillv docreases hv lliocoliisioiis till tlio 



stalioiiai-y stale is reached, ^^itll which, as appears, iMa\wI'',i,i, 's dislri- 

 hution of velocities exists. 



The second is the iiielhod half followed hy IJoTiTZJMANN, and entirely 

 by .Jkans, by which it is denionslrated thai on certain hypotheses 

 tlie state witii niiiforni density and thai with MaxwelTs disli-ibulion 

 of velocities are the most probable. 'J'hese hypotheses are, as regards 

 tlie distribution ol place, that every time lliere W'Onld be an e(|ual 

 chance to any place in the vessel for every molecnle; witii regard 

 to the distribntion of velocities that there would evei-y time be an 

 equal chance that the point of velocity of a molecule would get into 

 an}' arbitrarily chosen volume-element, in which we should finally 

 have to reckon with llie tact that the total energy has a certain 

 definite value. 



I have tried to show') that there is something contradictory in 

 this, Avhich might be avoided by assuming that the gasmass is 

 arbitrarily chosen from a microcanonical ensemble, of which all the 

 systems possess the energy wdiich the gasmass must have. For in this 

 all the combinations of place and all the combinations of velocities 

 with the same energv are equallv numerons, and so we ha\ e the 

 same chance to hit upon them for the system chosen. 



So another proof for the above mentioned result is furnished, when 

 we show that an arbitrary ensemble of systems with the same 

 energy, left to itself, passes into a microcanonical ensemble. Gn?Bs 

 endeavours to demonstrate this in the XIP'' Chapter of his "Statistical 

 Mechanics" ; the reasoning is made clearer by Lohkntz -), though the 

 latter goes no further than calling the assumption that we shonid 

 finally get a microcanonical distribution, very plausible. 



However in a recent paper '') Poincare called attention to a property 

 in the light of whicli, in my opinion, the aboxe reasoning is no 

 longer tenable. There Poinoare shows, namely, that the quantity 



S= i I^iog P<:Id\...(Lvn (in which j\ — r„ re[)resent the variables 



I) 



which determine e\erv svstem of a certain ensemble, and /^= — 



iV 



the coefficient of jn-obability, the integration being extended over the 



1) These Proc. Fcl)r. 21, 1906 and Jan. 24, 1007. 



") "ilber den zweilen Hauptsatz dcr Thermodynamik" ; Ahliandlangcn liher 

 Tlioorelische Physik, Leipzig 1906, p. 289. 



■■*) "Piéflcxions sui- la Tliéorie cinétique des gaz" ; Journal de Physique, 1906, 

 p. 369. 



