837. 



We shall first assume, that y(- > (liquid below the surface, at 

 least in the neighbourhood of the axis), as is the case in a wide 

 tube or with a drop lying on a surface. 



^ 2. When the meniscus is very large, it will be practically flat 

 near the axis and the curvature of the surface is only appreciable 

 near the edge. This naturally suggests, for the purpose of integrating 

 equation 1', dividiïig the curve into two portions: a central part, 

 where the angle if obtains but small values and which extends to 

 pretty near the edge, and a marginal part, where ff can assume 

 larger values aud which for smaller values of ip passes into the 

 central portion. 



Let / be a special value of x belongiug to the marginal part (for 

 which we shall take the abscissa of poiuf B in fig. 1). If / is suffi- 

 ciently large, it is clear that the marginal part of the curve cannot 

 differ much from what would be found in the two-dimensional 

 problem (/=oo); the width of the marginal part is then small as 

 compared to / (cf. ^ 3). whence putting x ■=z I -\- u, u. may be treated 

 as small with respect to P). For that part we may therefore write 



\ f u \ 



kz dz = sin (p d I -\- — sin <ƒ I 1 -\- . .. \dz, . , . (2) 



which equation may be solved by successive approximations. 



§ 3. To a first approximation we thus have, as in the two-dimen- 

 sional problem"), since z and (p become very small together*) and 

 u-=0 for (f = 71, 



of curvature at the top) is very large compared to x, i.e. that -Ro^^i ^ is a large 



2 



number and therefore /?1'^ + A; = — — -= a small number. 



Ro^dbk 



The figure is drawn on the assumption of R^ being positive (i.e. liquid below 

 for A; > and above for A; < 0) ; in the opposite case the meridional curve would 

 be obtained by turning the figure over about the .r-axis. 



1) This method of simplifying the problem was already used by PoissON, Nouvelle 

 théoiie de Taction capillaire. See also : A. Kerguson, Phil. Mag., (6), 25, (1913) p. 507. 



^) Compare for instance A. Winkelmann, Handbuch der Physik, 2e Aufl 1(2), 

 (1908) p. 1131. 



3) z does not become zero, however, for © = ; from which it would seem to 

 follow that equations (3) can only hold as long as f is not infinitely small. 

 On closer inspection they appear to remain valid (provided u<< h), since the 

 dependence of z on x for small values of z is of an exponential nature; indeed, 

 for (f small (3) gives m i^ /c = % (p + 2 — iogf 4 = /ogf Y' + 0,614. sKfc = rp = 



= 0,543 e"*^^* = 0,543 e^'^'^^ . C'^*^^"; hence the minimum-value which z attains at a 

 large distance from the edge is infinitely small of a higher order than the infinitely 

 small values of z in the neighbourhood of the edge 



