Capillary Surface inside a Tube of Small Radius. 135 



Thus, as is shown by (xxi.), all the formulae already obtained 

 may be employed, substituting H for h. 

 Putting 



p — ^ li x — li / = r) + H, . . . (xxii.) 



we require the conditions under which p is a maximum. 



du 

 Let tan (f> = ■—- , and let fa be the value of <£ for which 



x = r. Then tj and H are functions of fa, and the condition 

 for a maximum or minimum value of p is given by 



£=°- 



dfa 

 But, from (vii.) 



2a 2 . f 2 cos 2 0! sin 0! — 11 



xi= — sincpi— rcosecd)! i l + - . • 9 , - f 



r ^' r I 3 snr^ i 



and remembering that 



the required condition for a maximum value of p gives 



IT 



fa= - -, and therefore 



H -*"? r 



-LJ-max. — „ • » • 



r o 



Hence the radius of curvature at the vertex under maximum 

 conditions is given by 



Rss i af rBJ' • • • (xxiii - } 



and the corresponding value of 7) is 



A/3. , ■ s 



'= r --6T < xxlv -> 



The equation of equilibrium of the bubble under the con- 

 dition of maximum pressure is 



g ipjii-p'(h'+v)] =g-, 



and, substituting for K and rj from (xxiii.) and (xxiv.), and 

 reducing, we have 



where B is a known quantity. 



