﻿Virial of a Mixture of Ions. 565 



stating the "positions" of either the + or the — ions in it 

 (by the " position " is meant the number o£ the ion without 

 regard to sign counting outwards from A , i. <?., the suffix p 

 of A p ). Thus, the first arrangement illustrated above is 

 known when it is stated that the 



1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, nearest — ions are 



respectively the 



2nd, 4th, 5th, 6th, 11th, 13th, 14th, 15th, nearest ions to A , 



i. e., have the positions 2, 4, 5, &c. The probability of a 

 given arrangement must consequently be expressible as a 

 function of the positions of either of the + or of the — ions 

 in it. We shall take the central ion A as being -f in all 

 cases, and specify the arrangement by the positions of the 

 — ions, and transform (27) so that the probability of the 

 arrangement is given in terms of these. 



Let n be the number of — ions in the arrangement of the 

 in nearest ions to A , and let the positions of the 



1st, 2nd, 3rd, ... uh, ... nth 3 nearest — ions be respectively 



Pi, f2, P3, ,-,Pi, >--Pn. 



Then P(m) may be written as P(j> l5 p 2 , Pz, ... pi, ... »„), and 

 P(S) asP(l,3,5, ...2-/i—l). 



Now we can transform the given arrangement into the 

 standard one by transferring in succession each — ion from 

 its positioning •••/>»? to its standard position 1, 3, ... 2n — 1. 

 By carrying out these transferences in a suitable order it can 

 always be arranged that the order of succession of the — ions 

 is not affected at any stage of the process. Thus the order 

 of succession of the ith — ion will not be changed by the 

 transference (i. e., it will still remain the ith — ion) if all the 

 ions between pi and 2i — 1, including that at 2i— 1, are + . 

 If this condition holds, then the transference of this ion can 

 be effected without changing its order, and when this is 

 done there will necessarily be some other — ion for which the 

 same condition holds, and so on until all the transferences 

 have been effected. 



Suppose that the condition referred to above holds for the 

 ith — ion, which we shall call B l5 then we shall consider the 

 transference of B x from p { to 2i — 1 as being made as follows. 

 We first interchange B t at p { and the + ion next to it, and 

 we then repeat this process until B; has arrived at the position 

 2i— 1. Let p a be the position of B z - at any stage of the 

 transference, and p+ 1 that of the adjacent + ion with which 

 it is to be interchanged. We can state in terms of p and i 

 the values of the coefficients a v and a v+x of the terms which 



