﻿572 Dr. S. R. Milner on the 



value over all distances of q 2 /r v , or of the virial on A of the 

 vth nearest ion (assumed of positive sign), on a purely 

 random distribution of the ions. This can readily be proved 

 by direct integration, or it follows at once from (38) since 

 the exponential there reduces to 1 when the distribution is a 



m 

 random one. Hence 2 ±whu v is the mean virial of a par- 



ticular arrangement on a random distribution. The virial 

 contribution of a particular arrangement on a random dis- 



m 1 



tribution would be X ±whn l/ x ^, since l/2 m is the chance 



of this arrangement when the m nearest ions only are con- 

 sidered. Now in the distribution as modified by the inter- 

 ionic forces the contribution will be different from the above 

 for two reasons. First, the chance of the given arrangement 

 will be different — we have seen [(27)] that this is 



m 

 — h 2 opUp 



and, secondly, the mean virial of the arrangement will not 

 be exactly the same as it would be on a random distribution, 

 because the inter-ionic forces, as well as affecting the chance of 

 a given arrangement of signs, will also affectthe mean distances 

 of the ions. We shall here, however, neglecting the second 

 of these effects in comparison with the first, assume that the 



m 



mean virial of the arrangement is still given by 2 ±whu v in 



v—l 



the modified distribution. We shall then be able to write 

 for the virial contribution of the arrangement as an approxi- 

 mation to (38) 



m 



fUP(E)=2 ±»A«,x"C«~*^**- • (40) 



It is possible to show that the approximation is valid when 

 li is sufficiently small. (40) differs from (38) only in that 

 v p is replaced by u p . It follows from the expression for 

 v p that when v is moderately large (say greater than 10) 

 Vp 'becomes practically identical with u v . The virial contri- 

 bution of all except the first few nearest ions will, therefore, 

 be accurately represented by (40), and it is only in respect 

 to the average virials of these first few ions that any error 

 will be introduced by its use. Now it can be shown that as 

 h becomes smaller the average virial of these first few ions 

 becomes a smaller and smaller fraction of the whole, and the 



