146 MOTION OF SYSTEMS AND ENSEMBLES 



apply the processes of taking a limit. If treating the elements 

 of volume as constant, we continue the stirring indefinitely, 

 we get a uniform density, a result not affected by making the 

 elements as small as we choose ; but if treating the amount of 

 stirring as finite, we diminish indefinitely the elements of 

 volume, we get exactly the same distribution in density as 

 before the stirring, a result which is not affected by con- 

 tinuing the stirring as long as we choose. The question is 

 largely one of language and definition. One may perhaps be 

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

 the mean square of the density of the coloring matter, but an 

 infinite amount of stirring may be regarded as producing a 

 condition in which the mean square of the density has its 

 minimum value, and the density is uniform. We may cer- 

 tainly say that a sensibly uniform density of the colored com- 

 ponent may be produced by stirring. Whether the time 

 required for this result would be long or short depends upon 

 the nature of the motion given to the liquid, and the fineness 

 of our method of evaluating the density. 



All this may appear more distinctly if we consider a special 

 case of liquid motion. Let us imagine a cylindrical mass of 

 liquid of which one sector of 90 is black and the rest white. 

 Let it have a motion of rotation about the axis of the cylinder 

 in which the angular velocity is a function of the distance 

 from the axis. In the course of time the black and the white 

 parts would become drawn out into thin ribbons, which would 

 be wound spirally about the axis. The thickness of these rib- 

 bons would diminish without limit, and the liquid would there- 

 fore tend toward a state of perfect mixture of the black and 

 white portions. That is, in any given element of space, the 

 proportion of the black and white would approach 1 : 3 as a limit. 

 Yet after any finite time, the total volume would be divided 

 into two parts, one of which would consist of the white liquid 

 exclusively, and the other of the black exclusively. If the 

 coloring matter, instead of being distributed initially with a 

 uniform density throughout a section of the cylinder, were 

 distributed with a density represented by any arbitrary func- 



