256 Mr. S. H. Burbury 07i 



And for the total change of p with the tune for a particle 

 moving with the liquid 



dp 'dp , ^V A /ri \ 



or p is constant ^or the same particle for all time. That, I 

 think, is a rigorous conchision given the assumptions and the 

 definition of p, not merely an approximation. 



12. It follows from the proved constancy of p that 



\\] p^ da; dy dz, or p^^ is constant. 



Professor Gibbs then says, p. 145, " No fact is more familiar 

 to us than that stirring tends to bring the liquid to a state of 

 uniform mixture, or uniform densities of its components, 

 which is characterized by minimum values of the average 

 squares of those densities." There is then an apparent con- 

 tradiction between our mathematical conclusion that p'^ is 

 constant, and our familiar experience that p^ can be made to 

 vary. And " the contradiction,^' he says, "is to be traced to 

 the notion of the density of the colouring-matter, and the 

 process by which this quantity is evaluated." This " quantity," 

 he further says, p. 145, " is the limiting ratio of the quantity 

 of colouring-matter in an element to the volume of the ele- 

 ment," which is my Definition A. 



I quite agree with him as to the origin of the contra- 

 diction, but I think there is only one legitimate way of 

 avoiding it. and that is to chanoe the definition. The con- 

 elusion that he draws, p. 14(3, is, however, that " one might 

 be allowed to say that a finite amount of stirring will not 

 affect the mean square of the density of the colouring-matter, 

 but an infinite amount of stirring may be regarded as pro- 

 ducing a condition in which the mean square of the density 

 has its minimum value, and the density is uniform."" I 

 cannot persuade myself to accept this conclusion. I appeal 

 from Professor Gibbs the philosopher of Chapter XII. to 

 Professor Gibbs the mathematician of Chapter 1. 



13. But the term " density at a point P '^ may mean more 

 than one different thing, and may accordingly have more 

 than one different definition. We have, every one of us, a 

 rough general idea of what we mean by the term, though we 

 seldom take the trouble to express our idea in the accurate 

 language of mathematics. When asked to do so, we resort, 

 as it were by instinct, to Definition A, which has the advan- 

 tage of being mathematically irreproachable. Mathematically 

 irreproachable iudeed it is, but, for the present purpose, I 

 think, useless. To show that it is not the only possible 



