﻿Virial of a Mixture of Ions. 571 



where /<, u„ and a p are the quantities defined on p. 563, and 





-J) T(v) r(v-|) , ^ , 



\ X Tv n -Tv i\ wllcn ^"^ and 



The + in (38) is meant to indicate that the sign of the term 

 in the summation in v is to be taken + or — according as 

 the vth ion is + or — -. 



(38) when summed over all the possible orders of 

 succession in the signs of the ions (we will call this sum 

 X fUP(E)) will give the total average virial on A of the m 

 nearest ions. fUP(E) is the term contributed to this sum 

 corresponding to a particular arrangement of the signs of 

 the ions specified by the values of the a's, We will call it 

 for shortness the "virial contribution" of the arrangement. 



In order to obtain X f UP(E) we theoretically require to 

 proceed as follows. We must first of all transpose (38) £0 

 that the virial contribution is specified in terms of the 

 positions of the — ions in the arrangement. Then, taking 

 a single term of (38), say that corresponding to the vth ion, 

 we must sum it up for all the cases in which the vth ion 

 is +, i.e., for all the cases in which the 1st + ion is 

 the vth, the 2nd the vth, &c, and subtract the similarly 

 obtained sum of all the cases in which the vth ion is — . 

 The total average virial on A of all the m ions is then 

 obtained by giving to v the values from 1 to m in succession, 

 and adding the results. The symbolical expressions from 

 which these may be calculated for any finite number of ions 

 may be written down without difficulty, but the work of 

 petting arithmetical values from them is very laborious. 

 The reason of this is that the v P of (38) is really, as is seen 

 from (39), a function of v as well as of p, consequently 

 ihe functions which have to be summed take up a different 

 form for each value of v. If Vp were only independent of i», 

 the arithmetical calculation would be much simpler. In 

 order to obtain numerical values of the average virial, so 

 that a curve showing the way in which it varies with the 

 concentration of the ions can be plotted, I have contented 

 myself with a calculation from an approximation to (38) for 

 the virial contribution, which is explained and justified in 

 the following paragraph. 



Consider the meaning of the expression for the virial 

 contribution in more detail. +whu v represents the mean 



