188 Theory of a Non-Conservative Gas [CH. vm 



strength of each, given by an expression of the form of (453). Then the 

 strength of the family of clusters is MS. Comparing two families of clusters 

 we have already seen that the ratio of the two S'a will be infinite, so that 

 whether or not the ratio of the two M's is infinite, the ratio of the MS's is, 

 in general, infinite. Thus all the fluid belongs to a single family. It is 

 clear that systems in all the M clusters have the same statistical speci- 

 fication. 



224. Hence we may sum up as follows : 



Out of a number of systems which have a given value for the energy, 

 this value being sufficiently low, and have had their energy reduced from 

 higher values by dissipation, the systems meanwhile being unaffected by 

 external disturbance, all except an infinitesimal fraction will have the same 

 statistical specification. 



And as regards physical interpretation : 



If nothing is known about a gas except that it has cooled undisturbed from 

 some higher temperature, and is now at such a temperature that its changes 

 of state are not very rapid (by " very rapid " being meant comparable with 

 10~ 10 seconds), then it is infinitely probable that it will have a definite 

 statistical specification, which will depend solely on its present energy. 



The way in which this specification has, in theory, been arrived at, shews 

 how profoundly different it must be from that arrived at on the assumption 

 of conservation of energy, namely the specification which leads to equi- 

 partition. 



PRINCIPAL AND SUBSIDIARY TEMPERATURES. 

 Definition of Subsidiary Temperatures. 



225. The exact form of a law of distribution of coordinates possesses 

 interest in itself from the point of view of abstract mathematics, but as 

 regards utility it will be seen that its only importance is that it enables us to 

 calculate the mean values of different functions of the coordinates, averaged 

 throughout the gas. Of these functions of the coordinates, only one class of 

 function is of much importance, namely the class which represents the whole 

 or parts of the energy. 



Let us suppose, as in 77, that L, the kinetic energy of a single molecule, 

 expressed as a function of its coordinates, is reduced to a sum of squares in 

 the form 



21, = mu* + mv* + mw 2 + c^ + c^ + (456), 



where m is the mass of the molecule, u, v, w are as usual the velocities of 

 its centre of gravity in space, and rj 1} rj 2 ... are generalised momentoids. 

 Equation (456) expresses the kinetic energy as the sum of contributions from 



