94-96] Maxwell's Treatment of Equipartition 85 



assemblage of systems represented in a generalised space will all pass over 

 the same paths if allowed sufficient time, and hence the time-averages will 

 be the same for every system. Since the time-averages summed over all 

 systems are equal, it follows that the time-averages for each individual 

 system are equal. 



/ If, however, Maxwell's assumption is untrue and it must be borne in 

 mind that no single system has yet been discovered in. which it is not 

 untrue there seems to be nothing to be said in justification of deducing the 

 equality of the time-averages from the theorem of 90. The main point to 

 be noticed is that if the systems are subject to fortuitous disturbances, there 

 is no reason for supposing that a homogeneous distribution of density in the 

 generalised space (or, more generally, a distribution satisfying equation (203)) 

 will be permanent, or conversely that the permanent state will satisfy the 

 condition expressed by equation (203). And if this is not so, the attempted 

 extension to time-averages fails entirely. 



95. It may nevertheless be true that for fortuitous disturbances of a 

 special type the distribution expressed by equation (203) remains per- 

 manent, and it may also be that the converse is true, and that the only 

 permanent distribution is that represented by equation (203). I have 

 thought it worth while attempting to shew that this is actually the case 

 with a gas. 



Before attacking this question, one further observation must be made. 

 If our gas consists of an infinite number of molecules, we can select one 

 single molecule from the rest, and regard the remainder of the molecules as 

 the dynamical system, while the single molecule plays the part of the 

 fortuitous disturbing agency. The disturbances are not fortuitous in the true 

 sense, but since the single molecule collides only with an infinite number of 

 different molecules in turn, it might be legitimate to regard its action as 

 fortuitous. And again the energy of the system is not constant when the 

 single molecule has been removed from consideration, but it might be legiti- 

 mate to neglect the deviations of energy which are infinitesimal in comparison 

 with the whole. In the following sections, however, the argument is not 

 based on the somewhat doubtful foundations just sketched out : it rests on 

 the assumption of molecular chaos. 



Application to a g.as. 



96. We shall suppose a gas to be composed of a number of exactly 

 similar dynamical systems the molecules. We suppose that each molecule 

 is surrounded by a sphere of molecular action of diameter cr, these spheres, 

 as in 82, being of such a size that two molecules exert no action upon one 

 another except when their spheres intersect. 



