380 Prof. G. N. Antonoff on Interfacial 



In this case, we have at once, writing x = md, y = nd, 

 r = dsjm 2 + n 2 , the expression for the force 



ZmeH 2 2m 2 -3?i 2 

 d* ' (m 2 + n 2 )V 2 ' 



It is understood that if the lengths of the doublets have 

 different values (/ 1? Z 2 ), I 2 is replaced by l L l 2 in this formula. 

 The law is that of the inverse fourth power as already stated, 

 and as could in fact be shown at once from a consideration 

 of dimensions. 



When ??i = 0, we have practically a very simple case with 

 the doublets facing one another, the force in the perpen- 



dicular direction being of order -^. When n = 0, the 



doublets are in line, and at distance md apart, the force 



, . GeH 2 



becoming - — — - 



° {md)* 



For a given value of nd, a line of doublets is specified, and 

 the whole attraction of this line on one side of the original 

 doublet, is 



ZeH 2 ^ m(2m 2 ~3n 2 ) 



*de 2 l 2 f 2-3n 2 2(8- 



d* m+™ 2 ) 7/2 ( 4 



■>-3n 2 ) 3 (18 -3Q "I 



+ n 2 )V 2 * (9 + n 2 ) 7 / 2 + '"J" 



This series cannot be summed in a convenient manner, but 

 a sufficient approximation may be obtained by noticing that 

 each term decreases as n increases. When n = 0, the terms 

 are of order m~ 4 , and the second is only about 1/16 of the 

 first. Thereafter the convergence is very rapid, and it is 

 sufficient for our purpose to ignore all but the first two 

 terms. 



The bracket changes its sign when n—1. Thus its values 

 are effectively, for n = 0, 1, 2, 



16 



2 + 



47/2- 



= 2i=2-125 5 





~~%U2 + 



10 



57/2 - 



1 



8 V2 4 



2 



25 V5' 



10 



8 







57/2 



4 7 ' 







