53-55] The Method of General Dynamics 47 



fore clear that these results will not be affected by constant multipliers 

 or additive constants in the definition of K. Hence K might equally well 

 and more simply, have been defined by 



K= I !!v log vdxdydz (93). 



54. In whatever form K is defined, the minimum value of K is given by 

 equations (72), and this is equivalent to 



v = constant ................................. (94). 



Thus the result of 50 states that in the original generalised space, all 

 except an infinitesimal fraction of the whole space represents systems in 

 which v is constant throughout the gas. This is the result which we might 

 a priori have expected. For selecting a point at random in the generalised 

 space is equivalent to assigning positions to the molecules at random, and we 

 should in this case naturally expect the resulting gas to be of uniform density. 



When the gas is nearly, but not quite, of uniform density, let the density 

 at any point be denoted by v = v + 8v, then equation (92) becomes 



Hence K is proportional to the mean value of (8z>) 2 averaged through the gas, 

 and we see that K is a quantity which might naturally be taken to measure 

 divergence from homogeneity. 



The Partition of the Generalised Space Velocity Coordinates. 



55. We can investigate the partition of the velocity coordinates in a 

 way exactly similar to that in which we have investigated the partition of 

 the positional coordinates. The density v must be replaced by the density r 

 denned as in 13. If we introduce a quantity H defined by 



H = I I T log rdudvdw (95), 



we can prove, exactly as before, that for all except an infinitesimal fraction 

 of the whole series of systems represented in the generalised space, the 

 quantity H only differs infinitesimally from its minimum value. Since we 

 are not tied down as regards additive or multiplicative constants in the 

 definition of this function, we may replace equation (95) by 



(96), 



where / is the / of 13, so that H is now identical with the H of the 

 preceding chapter, defined by equation (13). 



