﻿558 Dr. S. R. Milner on the 



we can see that, so long as e is not actually zero, the first 

 of these integrals can always be made small compared with 

 the second by sufficiently increasing the value of R. When 

 this is the case, the error produced by treating the exponent 

 as small throughout will be only a second order quantity. 

 Now the problem with which we are concerned is exactly 

 similar to the above, except that there is a large number of 

 ions existing in a correspondingly larger volume, and an 

 exactly similar argument will show that our assumption can 

 always be made valid by sufficiently increasing the volume 

 in which the ions are contained. The assumption can con- 

 sequently be justified, but it is to be observed that it restricts 

 the validity of the result for the virial to dilute concentrations 

 of the ions. In the application to electrolytes this is equi- 

 valent to assuming that the concentration is so dilute that no 

 appreciable association of the ions exists — at any rate as the 

 result of the electric forces. 



Returning now to the summation of (11), we shall simplify 

 matters by choosing our units of distance and of energy so 

 that all distances are measured in terms of (V/2N)i (which 

 is the edge of a cube which on the average contains one ion) 

 as unit, and all energies in terms of w as unit. We can 

 then write unity for 2N/V and for w in (11), which then 

 becomes 



P(E) = k m e p=i i=o n P 3 dv l dv 2 . . . dv m . . (15) 



We have now, first, keeping the signs of the ions fixed, to 

 integrate (15) over all possible values of i\, v 2 -..v m , and 

 then to sum up the result for all possible arrangements of 

 the signs. 



Let #i<£i?\i, 0m<f> m r m be the polar coordinates of A b 



Am. (15) then becomes 



m p-lq t Qp 4 



P(E) = *„« p=u= : d<f> 1 sm0 1 d6 1 r 1 *dr 1 ...d<l> m sm0 m d0 m r m *dr m 



* ' ' < 16 ) 

 This has to be integrated with respect to each and cf> over 



the whole sphere having the corresponding r as radius. 



Consider the integration with respect to the 6 P and <f> p of a 

 single ion A p . Suppose that the integrations for all the ions 

 outside A Pt i. e. with respect to m , <j) m , &c, down to p +i, (f> P +i 

 have been already performed. It is evident that by these 

 integrations all the terms of the form g p q s /r ps , where s>p, 

 will have been removed from the exponent of e in (16), and 

 in integrating for Ap we shall be concerned only with terms 



