the Variation of Entropy, 255 



caution ; and he gives an illustration, by the study of which 

 the reader is invited to the conclusion that the quantity 77, 

 rigorously proved (as ai)pears to me) to be invariable with 

 the time, will nevertheless be variable if the time be only 

 long enough. I think we should exercise caution before 

 we accept that conclusion. Of course, if the constancy 

 of Tj depends on the fulfilment of certain conditions, it may 

 be that with lapse of time the conditions will fail, and 77 

 cease to be constant. But we are not told what conditions, 

 nor that any conditions, will fail. 



10. Let us then consider the illustration. It will appear that 

 it does point to a way of escape from the difficulty, but not, 

 I think, exactly the way that Professor Gibbs recommends. 

 An incompressible liquid hns colouring- matter distributed 

 continuously through it. If P be a point in the liquid, p, the 

 density of colouring-matter at P, shall be defined in a way 

 precisely analogous to the definition of D above given, 

 namely — Definition A — p at P is the quantity of colouring- 

 matter within a sphere of radius r described about P as centre, 



'iirr^ 

 divided by ^^ — , in the limit when r becomes indeiinitely 

 o 



small. Professor Gribb?, as we have seen, did not expressly 

 formulate this definition for D, but he does expressly for- 

 mulate it in the case of p (p. 145). 



Now let u^ V, IV denote the component velocities of the liquid 

 at any point, and consider the element of volume dx dy dz. 

 By the motion of the liquid colouring-matter is passing into 

 this element regarded as fixed in space through some faces, 

 and out of it throuo^h other faces, and in exact analoo-y to 

 equation (1) w^e have 



| = -{£(^")^|(''")-|(^'^')}- • 



But in exact analogy to (2) 



du dv dw _^ . 



dx dy dz V- / 



Therefore in exact analogy to (3) 



'dp { dp (lQ , dp\ 



dt \ dx dy dt J ^ ' 



Let'now ^— - denote the chano;e of p for a i)article movino- 



----,>'-,. o t o r L o 



with the liquid due to that motion alone. Then 



(la) 



