( 308 ) 



of density over the elements of extension indicated by A in tig. 2. 

 This tendency proceeded from two actions, the former giving uniform 

 density over the horizontal, the latter over the vertical regions. So 

 we might say that the decrease of H is brought about both by the 

 motion of the molecules (the iirst action) and by the collisions (the 

 second action). The coordinates and the velocities occur also in // 

 in the same way. 



Yet Boltzmann states expressly that the H can only decrease in 

 consequence of the collisions ') and he shows this as follows : 



The change of H within a given surface is determined by 



- 2 logf= I | do doj ——logjl §— -f- 17— -f g — + A— + 1 — + ^ 

 dt „ft J J \Jt • \ d.r dy dz d$ dr t d$J 



+ C A {l>ogf)+ C ê (hgf). 



If the surface is made to join the walls of the vessel, the first 

 term is zero, the terms with A, V, and Z are lost if we assume 

 that there are no external forces ; C\ and C s denote the change 

 caused by the collisions. The change in consequence of the motion 

 of the molecules is equal to : 



SI 



. df dr' df 

 dodu) log. /IsT + lr + Sr 



Boltzmann shows that this integral is equal to: 



Cl dtodS/N — ( I dwdS Nf log f 



which is to be integrated over the surface S, which includes t lie 

 considered gas mass. From this follows that the increase in H in 

 consequence of the motion is equal to the quantity that is brought 

 into the surface S by the molecules. So if the gas is left to itself, 

 this quantity will be zero, so that H does not change in consequence 

 of the motion, but only in consequence of the collisions 2 ). No doubt 

 we shall have to look for the explanation of this difference in result 

 to the fact that Boltzmann considers the "entropie tine", whereas 

 above the "entropie grossière" was considered. If the elements do 

 and (Uo are taken of finite size, as must be done here, the calcu- 

 lations which reduce the change in H to a surface integral, must 

 not be adopted in unmodified form. 



§ 3. For a kinetic derivation of the 2 nfl law of thermodynamics 

 it is necessary kinetically to detine a quantity which agrees in 



i) See: Vol. I, p. 126, note. 



~) Gf. Lohentz, "Abhandlungen fiber Theoretische Physik", Abbandl. VIII. 



