﻿552 Dr. S. R. Milner on the 



particular configuration of the ions, take for a pair of ions 

 the product of tbeir mutual force into the distance between 

 them, reckoning it + when the ions are like, and — when 

 they are unlike ; sum this quantity for every pair which can 

 be formed out of the 2N ions, we get the virial for this con- 

 figuration. Multiply this sum by the probability of the 

 occurrence of this configuration, and sum up for every pos- 

 sible configuration of the ions in the volume, and we get the 

 average virial. The virial here considered is that due to 

 the electrical forces only: it is reckoned + when repulsive, 

 and it is the same as the potential energy of the ions, if we 

 understand by this the work which the electrical forces are 

 capable oE performing as the ions are moved to infinite 

 distances apart. 



When the average virial is known, the pressure of a 

 gaseous mixture is easily determined by Clausius' theorem, 

 or in electrolytes the osmotic pressure by thermodynamical 

 reasoning, and the result has important applications to the 

 dissociation theory of electrolytes. 



If the ions were uncharged w T e should be justified in 

 assuming that their distribution in the volume V was a 

 random one, for we have no data for asserting it to be any- 

 thing else. (By a random distribution it is meant that on 

 taking an instantaneous view of the system any given ion is 

 just as likely to be found in any one place in the volume as 

 in any other.) If the distribution were a random one when 

 the ions are charged, the virial might be either -f or — for 

 any particular configuration, but the average over all cases 

 must necessarily be zero ; for the ions existing in any two 

 given positions are just as likely to be of like as of unlike 

 sign, and so the term which they contribute to the whole 

 sum is necessarily zero on the average*. The distribution 

 will, however, not be random when the ions are charged ; 

 the effect of the inter-ionic forces will be to increase the 

 chance of the occurrence of those configurations which have 

 a negative virial, and to diminish the chance of the occurrence 

 of those configurations which have a positive one. The 

 average virial is consequently negative. 



It has been proved by Boltzmann + (Vorlesungen uber 

 Gastheorie, 1898, Part it. p. 137) that if T(E), F'(E') 



* This is only strictly true when N is so large that the difference 

 between iS" and N — 1 is negligible. 



■j- It should be mentioned that the statement in the text represents a 

 slight extension of Boltzmann's actual theorem. Boltzmann stated the 

 theorem only for a " constellation " or group of molecules in the system, 

 but it seems to me there is no reason why we should not extend the 

 croup so as to comprise the whole system, if we so desire 



