209-212] General Dynamical Theory 179 



The hydrodynamical equation of continuity in the generalised space 

 (equation (130)) is 



, ........................ (432), 



Dt r s=i\dp g dq,J 



where, as before, D/Dt denotes differentiation following an element of fluid in 

 its motion. 



From equations (430) and (431) we obtain 



dp. , 3g. = W 



dp s dq s dp s dq s ' 



so that equation (432) becomes 



VP-'Z -**- (433) 



2* ',3 dp s dq s ' 



In the special case in which F=0, this reduces to the simple form 

 Dp/Dt = 0, which is, of course, the expression of Liouville's Theorem. 

 Equation (433) therefore expresses the general theorem which must replace 

 that of Liouville when the conservation of energy does not hold. 



211. In the former case, in which we had F=0, the procedure was one 

 of extreme simplicity. Since we had Dp/Dt = 0, it was clear that a distribu- 

 tion of fluid which was of uniform density at one instant would remain of 

 uniform density throughout all time. Hence a generalised space filled with 

 fluid of uniform density supplied a basis of investigation which was inde- 

 pendent of the time. It was therefore possible to prove, by taking a census 

 of the different points in the generalised space, that all except an infinitesimal 

 fraction of the space represented systems for which certain propositions of the 

 Kinetic Theory equipartition of energy, etc. were true, and hence that 

 these propositions would be true (except for a theoretical infinitesimal 

 probability of error) for a gas selected at random. 



In the present instance this method fails because the space filled initially 

 with homogeneous fluid does not supply a basis of calculation which is inde- 

 pendent of the time. It may be (and, as will be seen, actually is) the case 

 that the fluid tends to crowd just to those parts of the space which represent 

 systems in which the former propositions are not true. To investigate these 

 questions, we commence by seeking for a basis which shall be independent of 

 the time. 



212. For the sake of brevity, let us write 



s=n \z 



(434), 



s =i dp g dq s 

 so that we can write equation (433) in the form 



j\. r * (435). 



122 



