Capillary Surface inside a Tube of Small Radius. 12^ 



Remembering that y is to be assumed small in comparison 

 with h, we obtain, after integrating (i.) twice, neglecting y 

 in comparison with h, 



y = k- \Zk*-W*, (ii.) 



the equation to a circle of radius h (k= -7-). Putting this 



approximate value of y in (i.) and integrating once, we have 



Ji 2 + 2a 2 ( 4a 4 -fe 2 )t 8a 4 ... . 



Z ~ 2a 2 h X+ %a?h z a> "dhV ' ' ^ 



or, putting lix — 2a 2 sin 6, 



. n 2a 2 r3 sin 2 d + 2 cos 3 -2n ,. N 



, = sm + _^ __ J. . (lv .) 



O 2 



Since 0= — — , and ^U- is a small quantity, (iv.) may be 



\Jl-\-p 2 ,l 



integrated in series, giving finally 



_ 2a 2 \/ia±-K 2 x 2 8a 6 1 4,a l 



8a\ 2a 2 + s/±a"-li 2 x 2 , . 



+ 37? log Ia~ 2 ' ' W 



as a second approximation to the Cartesian equation of the 

 meridional curve of the capillary surface inside a small 

 capillary tube. 



§ 2. One of the constants of equation (v.) is A, the height 

 to which the liquid rises in the given capillary tube ; it is 

 convenient to obtain an expression for h in terms of the 

 radius of the tube, and the contact-angle i of the liquid with 

 the tube. 



In equation (iii.) z= sin c/>, and putting sin cf>= cos i when 

 x = r, (iii.) becomes 



rh r (4a 3 -AV)* _8a* 

 ~2a 2 + A + 3a 2 AV "~ 3/iV * (v1 ^ 



COS! 



Hence, approximately 



7 2a" 

 h= — cos 2, 

 r 



which value, substituted in the small terms of (vi.), gives 

 more accurately 



2a 2 . . f . , 2 /sin 2 i . cos e — 1\ I .... 

 *=— 00.1-T.eo.^l+gl ^ )j. (vn.) 



Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. K 



