﻿Virial of a Mixture of Ions. 573 



error produced by substituting u p for v v will become smaller 

 and smaller as h diminishes. 



Using the approximation (40) for the virial contribution 

 of the arrangement, we can adopt a simpler method of sum- 

 mation than that previously described, since the exponential, 

 being now independent of v, can be taken outside the 2, 

 thus 



TO 



m —AS aptlp 



fUP(E)=w7i(2 ±u v )xh m e r=i . 



If the arrangement contains n — ions, which have the positions 

 Pi ... p m then 



2 ±t«y = Mi + M 3 ... — Up x -{-Vp r + \ ... — Up 2 + &C. . . (41) 



Let S(m)==— u 1 + u 2 ^ W3 + W4— to w terms. . (42) 



ivh$(m) is the mean virial on A of the standard arrange- 

 ment of m alternate — and + ions. From (41) and (42) 

 we get 



m 



2 ±w„=S(m) + 2[(w 1 — 1^) + (w 3 — wp 2 ) + (u 6 — ^3) + & c -] 



n 



= S(m)+2 2 Ui(pi), 

 i=l 



where w»(jof) is written for shortness for u 2 i-\ — u pi . The 

 values of Ui(pi) may easily be calculated as desired from 

 the table of u p on p. 564. For example 



tt 3 (8) = M 6 — w 8 = 0-61298-0-51460 = 0-09838. 



We have also, by (37) and (27) 



m 



consequently 



f up(E)= W A[s(m) +2 2 m(>o] x ' l(Pl) ;;' ^ ( ^- 



All the possible orders of succession in the signs will be 

 obtained by summing with respect to p± from 1 to p 2 —l 

 inclusive, then with respect to p 2 from 2 to p 3 — 1, and so on, 

 finally with respect to p n from n to m. Now in these sum- 

 mations S(??i) is a constant, and can be taken outside the 

 2'?. Also each u^pi) is constant until that summation 

 which applies to pi has been reached. Taking these facts 



