﻿568 Dr. S. R. Milner on the 



of the numbers can be explained as follows : — Consider a 

 particular value, say e 3 (p) at p=8, or e j 3 (8). Take a large 

 number of instantaneous views of the central ion A and its 

 m nearest neighbours. Pick out all those views in which 

 there are the same number n of — ions present ; of these 

 pick out all the sets of views in which the 1st and 2nd, and 

 the 4th to the ?zth, nearest — ions are situated in any 

 definitely fixed positions, which may be arbitrarily chosen, 

 except for the stipulation that the 2nd — ion lies inside the 

 position 5 and the 4th outside 8. In this set there will be «*i 

 certain number of views in which the 3rd — ion occupies the 

 position 8 and a certain number in which it occupies the 

 position 5, which is the standard position for this ion. The 

 ratio of these two numbers is <? 3 (8) or '89676, and is the re- 

 lative probability of the occurrence of positions 8 and 5 for 

 the 3rd — ion (the stipulation above simply meaning that 

 both positions shall be possible ones) . 



If we apply the foregoing transformation to each of the 

 n — ions in succession, we see that we can write, as a com- 

 plete expression for the probability of the arrangement in 

 which the — ions have the positions p l9 . . . p m relative to 

 that in which they have the standard positions, 



Fjpi ...pt ...£>„) 

 P(l, 3, ...2n-l) 



ei{pi)...ei(pi)...e n (p a ) - - (34) 



where the functions ei(pi) are given by (33). The relative 

 probability of the arrangement, as it will be called (the 

 words "to the standard arrangement" being understood j, 

 is expressed by this equation completely in terms of the 

 positions of the — ions. 



Next consider the relative probability of an arrangement 

 in which the position of B 1} the nearest — ion, is unspecified, 

 while all the other — ions are in definite positions p 2 , ...p». 

 By considering instantaneous views we see that it may be 

 obtained by giving to (34) the different values which it takes 

 up in succession for each possible position p 1 of B ls i. e., for 

 the values of p i from 1 to p 2 —^- inclusive, and summing the 

 results. Since e 2 (p2) ••• e n (p n ) are constant in this summa- 

 tion they may be taken outside the 2 ; , and, writing them in 

 the reverse order to (34), we get for the relative probability 



en(p n ) e 2 (p 2 ) X *iO>i). . . . (35) 



The relative probability of an arrangement in which not 

 only the position of B l5 but also that of B 2 , the second 



