254 Mr. S. H. Burbury on 



D be the density in phase in which that system finds itself 

 for the time beino- we have a change of D with the time due 



to that motion alone. And denoting this by -^ , we have 



From (3) and (4) is deduced (p. 9) the theorem that the 

 total change of JD with the time for a moving system (call it 

 dl)\. 



dD s^/'dD ' dBs 



7U=^-{^dq^^'d]^) 



^/'dD ' dBxY . . . c (5) 



^^[^d^-'^'-d,;j 



= 0, 



or D remains constant for all time for the same moving 

 system. 



Subsequently Professor Gibbs writes (p. 16) 



iS beino- the whole number of systems^ and 77 is a function 

 afterwards used to express entropy. Evidently, since D is 

 constant^ rj is also constant for each system for all time. 



8. Professor Gibbs does not formulate in extenso the com- 

 plete definition of D. But having regard to the foregoing, 

 I shall not be misrepresenting him if I formulate it thus : — 



Definition A. — D is the number of systems within the 

 extension in phase dqi.-.djyn, or da^ divided by 6?o-, in the 

 limit when each of the factors dq dp which constitute da 

 becomes infinitely small. 



I think the result D = constant is not an a2)proximation at 

 all, but a rigorous deduction from the assumptions which 

 have been made, and the definition of D. Whether it be 

 consistent or not with the theory subsequently developed, 

 \\ e cannot, I think^ escape from it. except by changing the 

 definition of D. 



9. 1 now come to Chapter XII., '' On the Motion of 

 Svstems and Ensembles through Long Periods of Time.'''' 

 The object of this chapter is to show that, if time be only 

 long enough, D, and by consequence 77, may vary. Professor 

 Gibbs begins by calling attention to the fact that 7} has been 

 proved to be constant. But he says we must exercise 



