Statistical mechanics 39 



(64) . ^^=Y{f\)\ogf,ih,cU. 



Evidently the function H might equally well have been defined by the formula 



(65) H =j\ogf,.f,ch,dil, 



where only other denominations of the variables have been used. 

 Or we may write 



IT = f log // . // ds^' dO. 



or 



H = f log /; . dB, dÜ 



We thus obtain four different forms of the equation (64), where in turn the 

 following four expressions of the passage function are to be used: 



V (./, ) = j ds, dAh dh (// — /,) ho , 



V if, ) =/ ^^^1 d'^ dh if,'/,' -f,,Q ico, 



V (//) = / de,' d^y db' if, f\ - f; /;) , 



V (A') = / äe; d^h' d,h' [f,f, -//./;') h\o' . 



Substituting these expressions in (64), exchanging the variables with the help 

 of integral invariant, so that the same variables occur in all integrals, and addiug 

 the four thus obtained values of dH:dt, we get 



(65) _ ^ =-l^dÜdB,ds,d'!^db\og-^{f,' -f\f,)hi^. 



As 



at the same time as 



we infer that the right member of (65) is always negative. 



The H-f unction is always decreasing, except in the case f\' — fifi = 0, when 

 H is constant. 



This is the ^-theorem of Boltzmann. 



From the ^T-theorem the theorem of Maxwell regarding the final distribution 

 of the velocities may be derived. B^or this point I refer to the treatises on the 

 kinetic theory of the gases. 



30. The paradox of Loschmidt. 



Regarding the interpretation of the iZ-theorem the following remarks may 

 find place. 



