﻿Virial of a Mixture of Ions. 557 



in the intermediate formulae will not affect the final result, 

 since we ultimately take the limit of the virial on A of its 

 m neighbours for m=x> , and this is practically equivalent 

 to extending the group so as to include the whole of the 2N 

 ions in the volume, to which case we assume that Boltzmann's 

 theorem is applicable. 



Since P(E) contains the unknown constant k m , we shall 

 have to sum up P(E) as well as U.P(E), in order that, by 

 writing XP(E) = 1, km may be determined. We shall attack 

 the summation of P(E) first, because it is the more convenient 

 for explaining the notation &c. adopted. 



In carrying this out we shall make the assumption 

 that any single term in the exponent of (11) of the 

 type q 2 jrw may be treated as a small quantity over the 

 whole region throughout which the integration is to be 

 effected. The conditions in which this assumption is justi- 

 fiable will be best made clear by considering a simple case. 

 Suppose that we have two ions only, a positive one A fixed 

 at the centre of a sphere of radius R, and a negative one B, 

 which may be situated anywhere within the sphere, the 

 distribution being a random one except as modi tied by the 

 attraction between the ions. The chance that B is situated 

 at a distance from A between r and r + dr is by Boltzmann's 

 theorem (3) easily seen to be 



k^AnrfdrfeirB* (12) 



The unknown k in this expression is given by the fact that, 

 since B must lie within the sphere of radius II, and since 

 also it cannot come nearer to A than a certain minimum 

 distance of approach (say e, this will depend on the sizes of 

 AandB), 



j\/ /m 3^r/RB = l (13) 



What are the conditions in which the exponent <f/rw can 

 be treated throughout the whole region of this integration 

 as a small quantity ? At first sight it would seem that it 

 would be necessary to suppose that e must be sufficiently 

 large to make q 2 jrw small right down to the lower limit e of 

 r, but this is not really the case. For if we write r as the 

 smallest value of r at which q 2 jr'iv can be considered a small 

 quantity, by writing the integral in (13) in the form 



gj'V/»f^ + jjjfV* ,, *fo. > . . (14) 



