55-58] The Method of General Dynamics 49 



As in the case of the positional coordinates the law of distribution of 

 velocities could have been predicted from considerations of probability. 

 For selecting a point at random in the region corresponding to systems of 

 energy E is equivalent to assigning velocity coordinates u, v, w to the 

 molecules at random, subject only to the condition that their squares shall 

 be distributed about a certain mean-value. It is therefore natural to find 

 that the distribution of velocities should be in accordance with the law of 

 trial and error. 



This law, it will have already been noticed, gives f= when u, v or w is 

 infinite. There is therefore the a posteriori objection to the analysis by 

 which it has been obtained, that if we divide all possible velocities into 

 "cells" in the manner of 39, the number of molecules in some of these 

 cells cannot legitimately be treated as infinitely great. The difficulty is best 

 met by taking a definite velocity V such that those molecules of which the 

 velocities do not satisfy the inequalities 



u < V, v < V, w < V, 



form an infinitesimal fraction of the whole. If the velocities which satisfy 

 these inequalities can be partitioned into cells in the manner of 39, so as to 

 satisfy the condition that the number in each cell is very great, then there 

 is no further difficulty, and equation (102) will give the law of distribution of 

 velocities which are less than V. The law now has no meaning for velocities 

 greater than V. It is obvious, for instance, that the law expressed by 

 equation (102) does not impose any upper limit whatever on the possible 

 values of u, v and w for a single molecule, whereas in point of fact such a 

 limit is definitely imposed by the energy equation. 



Molecules of finite size. 



57. It is obvious that it is in no way material to the analysis of 

 55 and 56, from which the law of distribution of velocities is found, 

 whether the regions mentioned in 33 are excluded from the generalised 

 space or not. For the exclusion of these regions affects the velocity 

 coordinates equally throughout. Thus the law of distribution of velocities 

 is the same whether the spheres are of finite size or are infinitesimal. It 

 remains the same right up to the extreme limiting case in which the spheres 

 are packed so tightly in the containing vessel that they cannot move. 



The Normal State. 



58. In the last chapter it was found, with the help of the unwarranted 

 assumption of molecular chaos ( 15), that the law of distribution expressed by 

 equation (102) represented a "steady state" for the gas. In the present 

 chapter it has been shewn, without making any use of this assumption, that, 



j. 4 



