﻿574 Dr. S. R. Miliier on the 



into account, and putting the 2 signs in their proper places 

 so that each acts only on what comes to its right, we get for 

 the complete average virial on A , U n (m), of all arrangements 

 of the m nearest ions, which contain among them n — ions, 



w J t r *» P2- 1 



U "(' ?l ) = 77^T S (™)2 e u (p H )... 2 e^pu 



n f * i>i+2-l Pi+1-I p t — 1 p. 2 -l ^ -, 



+ 2 X \ 2 *„(/>„) ... 2 ft+i(p«+i) 2 ei(pi)ui(pd 2 ^-ife-i) ... 2 <?!(/>!) > 



Now the first of the terms in the brackets is, by (36), 

 S(»n) x fn(m), and denoting the remaining term by 2g a (in), 

 we have 



U^)=^[s(m)+2gg]. . . . (43) 



Using the notation of (36) for g n {m) also, we can write it 



n { m Pi+l — l -> 



;/„(m)= 2 i %e„(p n )... % «(p0m(j»,)./U(p»- 1) L 



1=1 y.pn—11 pi=i J 



and we see from this, by putting z= 1, 2, 3, ... w, in succession 

 in it, that 



gjm) = 



m r- Pll—l { 



2 ejjpn) \un(p«)f»-l(Pn-l)+ 2 0»-i(pb_i) j M»_l(p»-l)/»-a(pi.-l— I) +2 



+ 5 ^(pjcufW/ifft-m-T^OMifrO] ...11, 



.... (44) 

 in which each 2 applies to everything that comes after it. 

 From this equation the numerical value of g n (m) for any 

 finite values of n and m may be calculated from the tables 

 of e n (m) and f n (m) previously given, by successive operations 

 of a similar type to those used in forming the table £or/ n (m). 

 Table IV. gives a series of values of g n (m) for 7i = "3. "The 

 construction of the table may be most simply explained bv 

 noting that (44) shows that the values must satisfy the 

 following difference equation : — 



gn{m)=gn(m-l) + e n (m){g n _ l (m-l)^u n (m)/ n _ l ( m -l)}. 



For example (see Tables II. & III,, and for u 3 (S) p. 573) 



£3(8) = <7 3 (?) + e 3 (S){g 2 (7) + u 3 (8)f 2 (?) }, 



or 14'803 = 8-1149 + -S9676{6--1352 + '09838 x 13-4489}. 



