﻿Vlrial of a Mixture of Jons. 559 



of: the form giq p jr ip ^ where i<p, i. e., with the ions inside tlm 

 sphere r p . Write E p for the mutual potential energy ot* 

 A P with respect to all the ions inside its sphere, so that 



Vp flp fp-l,p i — r ip 



Then (16), so far as the integration in 9 P and <p p is concerned, 

 may be written 



P 1 (E)=f 7r f 2 ' Ce~ Ep dcf>^mOp(W P) . (17) 



where C is a factor which includes all the quantities which 

 do not vary with 6 p and (f> p . 



Now it is important to observe that we cannot treat E p in 

 (17) as a small quantity*, even though we have assumed that 

 each individual term of it, gig p /ri P , may be treated as small ; 

 for we shall have to consider the probability of cases in 

 which all the ions inside A/s sphere have the same sign, 

 and in such cases E P may be large if the number of ions 

 inside is large. But it will follow from our assumption that, 

 if E p stands for the mean value of the mutual potential 

 energy of A p and the ions inside its sphere which is obtained 

 when A p may be situated anywhere indifferently on the 

 surface of its sphere, then we can treat Ep — ~E p as a small 

 quantity. For E p at any point on the surface of the sphere 

 can only differ from its mean value over the whole surface 

 by more than a small quantity as the result of two causes. 

 First, there might be a very unsymmetrical arrangement 

 (considered with regard to sign) of a large number of ions 

 inside A p s sphere; for example, if all the positive ions were 

 in one hemisphere, and all the negatives in the other, then 

 E ? — E p would not necessarily be small at every point of the 

 surface. It is necessary that the number of ions should be 

 large, for otherwise both Ep and E« would themselves be 



£V,, ^l VU x*v»...iuw ~~„ lx ^y UU v. ^ p 



small, and their difference still more so. But it is easily 

 seen that as the number of ions is increased, the probability 

 of an unsymmetrical distribution of such a type as would 

 make E/> — E P a large quantity, becomes smaller and smaller 

 compared with that of a more symmetrical distribution such 

 as would not produce the effect. Consequently, when the 

 complete summation of (16) with respect to the positions 

 and signs of all the ions has been effected, the total con- 

 tribution to the sum due to these cases will be but a small 



* The attempt to treat it as small leads to impossible results for the 

 virial. 



