590 Mr. S. H. Burbury on the 



shall be F(U V W)dV dV dW, or shortly F dV dV dW.. 

 We may call these spheres of the class F. 



Similarly fdudvdw is the number per unit of volume of 

 spheres of mass m, whose velocities lie between the limits 



w . . u+'du ~) 

 v . . v + dv > a. 

 io . . w + dw j 



We may call these spheres of the class f. 



Boltzmann now assumes that the number of collisions 

 which take place per unit of volume and time between 

 spheres of class F and spheres of class f and in which the 

 coordinates defining the relative position are within defined 

 limits da, is 



F/R dJJ dV dW du dv dw da, 

 where 



E2 = (U- M ) 2 + (V-t>) 2 +(W-w)*. 



That is, he assumes that the chance of a sphere M having 

 velocities within the limits A is independent of 1 lie position and 

 of the velocities of the sphere m,, howevtr near the two spheres 

 may be to one another. This we may call the condition of 

 independence. It is assumed by necessary implication for 

 all pairs of molecules approaching collision with each other. 



12. Having made or implied this assumption, Boltzmann 

 from this point onward works rigorously. The truth of his 

 result depends on the truth of the above assumption. 



By n collision of the kind last described the two spheres 

 pass respectively into the classes F' and f, the numbers of 

 which are FWrfVW and f'du'dv'dio'. And by a process 

 so well known that I need not here set it out, he deduces the 

 H theorem. According to this theorem 



7TT +CO Tf /• 



-^ = M ( F !/'-*y) R % Yf du dv dw dv dY <* w > 



which is necessarily negative if not zero, and then only zero 

 when F[/' = F/' for all cases in which a pair of spheres can 

 pass by collision from the classes F/ to the classes F'/' or 

 vice versa. And the solution of the equation F'/' = F/ 

 involves, if there be no stream-velocity, 



whence we deduce 



ra^ 2 = MTP, &c. 



13. The result thus proved by Boltzmann, if his funda- 

 mental assumption is true, is that the motion is irreversible 



