Manchester Memoirs, Vol. Ixv. (1921) No. 4 9 



failure of the equations outside these limits affords no reason for scepti- 

 cism as to their reliability inside the limits. 



We shall now proceed to develop in as simple a manner as possible 

 the equation of the meridional curve of the surface inside a small vertical 

 capillary tube ; the argument will be given for a liquid of zero contact 

 angle — its extension to a liquid of finite contact angle presents no special 

 difficulty. 



Let Fig. 1 b represent the liquid in its tube, and Fig. 1 ^an enlarged dia- 

 gram of the surface. With axes OX and OY as shown the pressure-excess 



on the two sides of the surface at P is given (i) by T( =?- + =- j and (ii) by 



T 



gp(y + h). Equating these values, we have, where a 2 = — 



1 1 y + h 



Now R x is the radius of curvature at P of the curve AOB, and R 2 is the 

 length PN of the normal at P intercepted by the axis. Hence the above 

 equation becomes 1 



1 d ( • n\ y + h r\ 



--(*sin0)=-^ .... (1) 



Innocent as this equation may appear, its exact solution is, apparently, 

 impossible. Approximate solutions have been given by many analysts 2 

 by more or less complex methods ; I shall here attempt to outline a pro- 

 cess which will give us the second order correction in a relatively simple 

 way. 



In principle the method consists in working out an approximate value 

 for y ; this approximate value, substituted in equation (i), now makes it 

 immediately integrable. Integrating, we develop a more accurate value 



1 For (see any text-book on the differential calculus), 

 And, from Fig. 1 c 



Hence 



Ri 



1 de de dx de 



R a ds dx ' ds dx 



e. 



1 sin e 





R 2 x 





+ — = -( x cos e + sin e 

 R 9 x\ dx j 



) 



1/ d (sin 6) . „ 



= - *— ^ + sin e 



x\ dx 



dx\ 

 dx) 



1 d{x sin 6) 





x dx 





2 Laplace, " Mecanique Celeste," Supp. au Xe Livre (1805). Poisson, " Nouvelle 

 Theorie de l'Action Capillaire (1831), Chap. IV. Mathieu, " Theorie de la Capillarite " 

 1(1883), Chap. II. Rayleigh, Proc. Roy. Soc. (A), 92, 184 (1915). Lohnstein, Wied. 

 Ann., 54, 713 (1895). 



