﻿554 Dr. S. R. Milner on the 



Since 2N is a very large number this is practically identical 

 with 



e-~^\ (4) 



so long as v is not a large fraction of the whole volume. 

 The, chance that there does exist an ion in v is 



"(H) 



which reduces to 



2N 



(5) 



if v is infinitesimally small. 



Out of a large number of instantaneous views (or " cases ") 

 of the given ion A and its neighbours, the fraction of the 

 cases in which the nearest ion is situated in the volume dv 1 

 at the distance ?\ away from A is given by the chance that 



there is no ion in the sphere r.irri described round A as 



centre, combined with the chance that there is an ion in the 

 given volume di^. We get for this, from (4) and (5), and 

 neglecting the volume of the ion A in comparison with 



4 3 



2N " 4 , 9N 



s-v-s^.^i (6) 



The fraction of these cases in which the second nearest 

 ion A 2 is situated in dv 2 at distance r 2 from A , is the chance 



(4 4 \ 



g77r 2 3 — yTT^i 3 ) between A x 



and A 2 , combined with the chance that there is an ion in dv 2j 

 that is 



2N/4 . 4^ ? 



OT/4 ,_4 x 2N 



V \3 2 3 ' ) .-^dv 2 .... (7) 



Similarly for the third, fourth, fifth, nearest ions, A 3 , A 4 , A 5 , 

 right up to the with, so long as m is not comparable with 2N. 

 Multiplying together (6), (7), &c, we get for the probability 

 that the nearest ion to a given ion lies in a given elementary 



volume di\ at r x , the 2nd nearest in dv 2 at r 2 , the mth 



nearest in dv m at r m , 



p -f -|«w' 2N, 2N. 2N, ... 



