﻿556 Dr. S, R. Milner on the 



It will be observed that in equation (11) what is stated is 

 not the probability of a given configuration of the whole of 

 the 2N ions, but only that of a certain number m of ions 

 nearest to an arbitrarily chosen one A . The strict theo- 

 retical method of solving the problem would be, as has 

 already been mentioned, to calculate P(E) and E for a 

 given configuration of the whole of the 2N ions; the average 

 virial E would then be obtained by summing up the quantity 

 E.P(E) for every possible configuration of the ions in the 

 volume V. Instead of doing this we shall adopt a simpler 

 process which gives the same thing in the end. We first 

 determine the average virial on the ion A of all the m 

 nearest ions to it, the average being taken over all possible 

 positions and signs of each of these ; i. e., we sum up the 

 quantity U.P(E) for all possible configurations of the m 

 ions. If we can obtain the limiting value U, which this 

 approaches as m becomes infinite, we have the average virial 

 on A of all the remaining ions in the volume. We can 

 then write the total average virial of all the 2N ions in the 

 volume in the form iii = N U (the 2 disappears since the virial 

 of each pair of ions is only to be reckoned cncej. 



A certain difficulty with regard to equation (.11) may with 

 advantage be discussed here. Boltzmann, in deducing the 

 theorem on which (11) is founded, first proves it for the case 

 of two molecules close together under the condition that the 

 forces exerted between them and the remaining molecules 

 are negligible, and he then points out that the theorem may 

 be extended to any number of molecules close together under 

 the same conditions, Now in the case of ions, where the 

 forces vary comparatively slowly with the distance, the forces 

 between a group of a finite number of nearest ions and the 

 remaining ions in the volume will not be in general negli- 

 gible. Hence, while the theorem is doubtless valid for the 

 whole of the 2N ions taken together, it is open to question 

 whether it is justifiable to apply it in the way in which it 

 has been here done to a group consisting of a finite number 

 of nearest ions only. If it is not, then (11) must be looked 

 on as merely an approximation to the true expression. I 

 think it is probable, indeed, that (11) is only an approxima- 

 tion when m is finite, for the reason that, by the presence in 



2N 4 3 

 it of the factor e V 3 m ? the assumption is virtually 

 contained in it that the ions external to the group are 

 arranged at random, and this is not actually the case. It 

 may, however, be observed that any error due to this cause 



