the V\incUion of Entropy. 257 



definition, it is sufficient to substitute for the continuous 

 liquid of Professor Gibbs' illustration, a medium consisting 

 of discrete molecules, each o£ finite dimensions, so that if a 

 space ^ be small enough not more than one molecule can 

 have its centre therein at the same time. As applied to such 

 a medium Definition A is. not only useless, but wholly devoid 

 of meaning. The " number of molecules within dS when dS 

 becomes infinitely small " has no meaning whatever. 



14. Nevertheless the term "'density" as applied to this 

 system of molecules has a meaning capable of definition. 

 When we speak of the density at a point P, we evideutlv 

 contemplate many molecules, but we should say that mole- 

 cules very distant from P have nothing to do with the densitv 

 at P. For a complete definition we should perhaps weight 

 the molecules according to their nearness to P. But as this 

 would be difiicult, we might, as a rough makeshift, say p is 

 the number of molecules within a sphere of defined radius r 

 described about P as centre. That is unambiguous, and 

 though open to certain objections, for some purposes sufficient. 

 Returning now to our liquid we might define p the density 

 of colouring-matter at a point P in the liquid, to be the 

 quantity of colouring-matter withinr-a sphere of a certain 



477/-^ 



finite radius /' described about P as centre divided bv — ;; — , 



b 



or taking the volume of this sphere. for the unit of volume. 

 p is the quantity of colouring-matter in unit of volume at P. 

 The proof of the constancy of p now fails. In the same way 

 we might take a finite extension in phase including certain 

 values q. p, of the variables in Professor Gibbs' ensemble. 

 and define D as the number of systems in that finite extension 

 in phase divided by the extension in phase. The proof of the 

 constancy of D now fails. D and r) will vary, and there 

 is no necessity to resort to long periods of time. 



15. But assuming the variability of D. or rj. to be now bv 

 whatever means established, I find yet another difficulty in 

 the consequences which Professor Gibbs deduces from it. So 

 long as 7] remains constant for the same system, we mav 

 define rj to be the entropy which that system has. That is 

 not inconsistent with the definition of t] above given: D=X^''. 

 But 7] being now supposed variable for the same system, we 

 require a definition. It is (p. 148) " an arbitrary function of 

 the phase "^ in which the system finds itself. We are then 

 to consider a certain finite extension in phase V which might 

 be represented graphically by an area on the paper. This is 

 divided into elements of extension in phase denoted by 



Phfl, Mao. B. (). Vol. C. No. 32. Aw,. 1903. S 



