THROUGH LONG PERIODS OF TIME. 145 



value of the average square of the density of the coloring 

 matter. Let us suppose, however, that the coloring matter is 

 distributed with a variable density. If we give the liquid any 

 motion whatever, subject only to the hydrodynamic law of 

 incompressibility, it may be a steady flux, or it may vary 

 with the time, the density of the coloring matter at any 

 same point of the liquid will be unchanged, and the average 

 square of this density will therefore be unchanged. Yet no 

 fact is more familiar to us than that stirring tends to bring a 

 liquid to a state of uniform mixture, or uniform densities of 

 its components, which is characterized by minimum values 

 of the average squares of these densities. It is quite true that 

 in the physical experiment the result is hastened by the 

 process of diffusion, but the result is evidently not dependent 

 on that process. 



The contradiction is to be traced to the notion of the density 

 of the coloring matter, and the process by which this quantity 

 is evaluated. This quantity is the limiting ratio of the 

 quantity of the coloring matter in an element of space to the 

 volume of that element. Now if we should take for our ele- 

 ments of volume, after any amount of stirring, the spaces 

 occupied by the same portions of the liquid which originally 

 occupied any given system of elements of volume, the densi- 

 ties of the coloring matter, thus estimated, would be identical 

 with the original densities as determined by the given system 

 of elements of volume. Moreover, if at the end of any finite 

 amount of stirring we should take our elements of volume in 

 any ordinary form but sufficiently small, the. average square 

 of the density of the coloring matter, as determined by such 

 element of volume, would approximate to any required degree 

 to its value before the stirring. But if we take any element 

 of space of fixed position and dimensions, we may continue 

 the stirring so long that the densities of the colored liquid 

 estimated for these fixed elements will approach a uniform 

 limit, viz.', that of perfect mixture. 



The case is evidently one of those in which the limit of a 

 limit has different values, according to the order in which we 



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