Shape of Large Bubbles and Drops. 515 



already found. Thus from equation (ii. a) we have 



\/l+p 2 2d 2 6a 2 r 3r 



assuming, to avoid cumbersome algebra, that the radius o£ 

 curvature at the vertex is infinite. Solving this equation for 

 -y/l-hj/, and squaring, we obtain, after some reductions, 



,_ y , (4a , -y) [\ 8a 4 {8a 3 -(4a 2 -77 2 ) f } n 

 r~ (2a 2 -y 2 ) 2 L 3r/(2a 2 -^X4a 2 -*/ 2 )J> 



neglecting terms involving r~ 2 and higher negative powers 

 of r. This gives at once, to the same order of approxi- 

 mation, 



dx _ 1 _ 2a 2 -/ r 32a 7 -4a 4 (4a 2 -y 2 )* 1 



^~p"yv / ia^ 2 L + 3r/(2a 2 -^)(4a 2 -?/ 2 )J' 



whence we have 



r , 2a?-y 2 32a 7 4a 4 ^ 



which, putting 



y = 2a sin #, 

 gives 



sin J 3rJ sm 3 6 cos- # dry* 



This gives at once 



y 9 a 4 



^ G = a]0 -2a+V^-f WAa2 -^ + fry 



a 2 r2aV4a 2 ^p-ra (% 2 -4a 2 ) 3 ^ y - 1 



3rL y 2 \/±a 2 -y 2 2 ° 2a + \/4a 2 - y*J ' 



which, as it should do, reduces, when r is infinite, to the 

 usual expression for the equation to the capillary curve 

 formed by the contact of a liquid with a plane wall. 

 ££ The integration constant is determined from the condition 

 x=0 when y = q ; and substituting in the small terms for q 

 its approximate value a \/2, we have finally 



y 2a + \/±a 2 — q 2 , /r-= : 2 /r-= 2 



* = alog^. /r ^-^ +V4a 2 -?/ 2 -\/4a 2 - g 2 



<? 2a + v4a 2 — y 2 



a 2 paV 4a 2 -y 2 + a(3/ - 4a 2 ) _ y/2 + l ^ loo .JL 2 + \/ 2 _-i 



37- L y V4a 2 -# 2 \/2 + 2 ° g \/2 ' 2a 4- v / 4a 2 — y 2 J * 



2N2 



