ON OUR KNOWLEDGE OF THERMODYNAMICS. 91 



Law at any rate. For, taking 3/2A as the measure of the temperature, 

 we see that it is — 



(i) The mean kinetic energy of translation per^Vee molecule, 

 (ii) The mean kinetic energy of translation of the molecules in an 



encounter for any given configuration of the system, 

 (iii) The mean kinetic energy of the cm. of two or more encountering 

 molecules found by supposing their whole mass concentrated 

 at this CM. 



If the volume of the gas be increased, the number of encountering mole- 

 cules will be decreased, but their mean kinetic energy will still be equal 

 to that of the free molecules. 



51. The proof of the Boltzmann-Maxwell Law thus presents little 

 difficulty when collisions are replaced by encounters lasting only a limited 

 time, so that the molecules are sometimes free and sometimes in the 

 process of an encounter, even though these encounters be multiple. But 

 it fails when the molecules act on one another at all distances, because we 

 cannot then consider any group of molecules apart from the rest. More- 

 over, unless collisions or encounters take place to a certain extent indis- 

 criminately between molecules, the frequency of distribution for any 

 particular molecule may depend on other circumstances besides its actual 

 state, and the assumptions made in proving the equation F/=F'/' (§ 39) 

 are no longer necessarily true. Hence, none of the above arguments now 

 afford any evidence that in such cases the Boltzmann-Maxwell distribu- 

 tion is the distribution which a gas naturally tends to assume, even though 

 the possibility of such a distribution may not be capable of disproof. 



Thus, in the test case of molecules attracting one another according to 

 the law of the direct distance, they will if initially arranged according 

 to the Boltzmann-Maxwell Law remain so distributed ; but this law of 

 permanent distribution is not unique, nor is there any tendency among 

 the molecules to attain this law. 



In dealing with such forces as that due to gravitation, as in the case 

 of a gaseous nebula held together by the attraction of its parts, it is clear 

 that the attraction of the more distant portions of the gas can be repre- 

 sented by a field of external force whose potential is the gravitation 

 potential of the mass. Thus no difficulty will be introduced into the proof 

 of the Boltzmami-Maxwell Law, except when we come to take account of 

 the attractions of those molecules that are very near any given molecule. 

 Theseare probably feeble, except in an encounter. The problem, however, 

 requires fuller treatment than can be given in the space of this Report. 



Section III. — The Boltzmann-Maxwell Law consideked in 

 Relation to other Theories. 



The Connection with the Theory of Probability. 



52. The application of the Theory of Probability to the determination 

 of the law of distribution among gas-molecules forms the subject of several 

 very interesting and suggestive papers in the hands of Boltzmann and 

 Burbury. 



I have ventured, on my own responsibility, to introduce the well- 

 known terms ' a priori ' a.nd ' a posteriori probability ' in the following 



