﻿Liquid Drop suspended in another Liquid. 153 



The difficulty of solution arises from the fact that the right 

 hand expression in (2) is a function of y and not of x (as is 

 the case in Lord Rayleigh's paper). 



However, remembering that the curve is symmetrical with 

 regard to the axes, we may write 



7 9 9 . ™* , 



where a. and I are to be found by approximation from (2). 



JE. g., the approximately elliptical form discovered above 

 can be found by using as the first approximation 



b 2 —y 2 = ocx 2 , 



where a ( = I — e 2 ) is a ratio slightly less than unity. 



After substitution and an integration, (2) then leads to 



dy _ x ux z 

 ds ~ Hi Wc 2 ' 



no constant being required. 



a i dx u dy 



Also - =-^-y, 



as ocx as 



and, therefore, since 

 we obtain 



/ 1- . OCX 2 \ 2 / a b 2 — aX 2 \ -, ,.,x 



On expanding and equating the constant term to unity, 

 and the coefficient of x 2 to zero, we find that a is given by 







«(l-«) = 



/> 3 

 = 4W 



,2_ 





6" 



2/ 





"(1 



-* 2 )4/tc 2 



4/tc 2 ' 



or 



as before. 



It will also appear that with a sufficiently small value of 

 £ 2 or 1— a, that the terms on the left of (3) which involve 

 «z? 4 and x 6 are negligible. 



LJ.<j., the term in x l turns out to be 



1 ." .,* 

 64 ' /re 4 

 Its maximum value is 



3a 4 i 2 /64/iV or ** . e 4 . 



