﻿(20) 



1 Iria I of a Mixture of Ion s. 5 G 1 



When we have integrated this expression over all the 

 possible values of the r's we shall have obtained the probability 

 of a specified order of succession in the signs of the m nearest 

 ions. Call this P(m), as it is a function of the number of 

 nearest ions considered. Now, when we cany out these 

 integrations in the order r l ......r m in the first' integration, 



r x , being the distance from A of the nearest ion, may have 

 any value between e, which is the nearest distance to which 

 the ions can approach, and r 2l the distance of the second 

 nearest ion ; then r 2 may be given any value between e and 

 r 3 and so on, finally r m may have any value between e and 



m 

 — 2 a P/ r P 



infinity. ^ Hence, writing e P =i fully, so that each 



factor of it may come after the first integral sign which acts 

 upon it, we have for the integral to be evaluated 



f» 4 3 a m r rin _<^n 1 



P(m) = M e-s^-r mr jdr m \ e '— i v m ^ *dr m ^ ... 



* ■ r m =e J r m -l = e 



rr 3 _ <h fr. 2 _ °x 



1 e r 2 r 2 2 dr 2 1 e 7 \ r?dr Y . 



Now so far as the integration in r { is concerned, we can 

 in accordance with our assumption treat ai/vi as a small 

 quantity. But in general a p /r p cannot be considered a small 

 quantity ; a p is by (19) proportional to the total charge 

 inside A P 's sphere, and when p is large, as has already been 

 pointed out, this may become considerable if the ions inside 

 have all the same sign. The successive integrations may 

 nevertheless be carried out with a close approximation by 

 the use of the following approximate theorem. 



By integration by parts, and assuming that the term con- 

 taining the lower limit is negligible, we have 



C nr' n(n — 1) r' z ) 



n , approximately. • . (21) 



In order for this approximation to hold it is not necessary 

 that a/r, but only that a/m\ should be small. Now, in 

 applying this theorem to the successive integrations of (20), 

 the integrands remain of the same type throughout, and it is 

 easy to see that the exponent of r Pi i. e., the n of (21), always 

 increases at a greater rate than the value of a P in the 

 Phil. Mag. S.^6. Vol. 23. No. 136. April 1012. 2 P 



r« 



a 



r r 1l dr- 



/n-fl 



r 

 n+1 



a 

 r' 







/n+1 

 r 



e 



n + 1 



a 

 r' 



