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1915]                   'ON  THE  THEOEY -OF THE  CAPILLARY  TUBE                       351
'experiments by Richards and Coombs led to the conclusion that in the case of water the diameter of the wide tube should exceed 33 mm., and that probably 38 mm. suffices. Such smaller diameters as are often employed (20 mm.) involve very appreciable error. Here, again, we should naturally look to mathematics to supply, the desired information. The case of a straight wall, making the problem two-dimensional, is easy*, but that of the circular wall is much more complicated.
Some drawings (from theory) given by Kelvin, figs. 24, 26, 28 f, indicate clearly that diameters of 1'8 cm. and 2'6 cm. are quite inadequate. I have attempted below an analytical solution, based upon the assumption that the necessary diameter is large, as it will be, if the prescribed error at the axis is small enough. Although this assumption is scarcely justified in practice, the calculation indicates that a diameter of 47 cm. may not be too large.
As Richards and Coombs remark, .the observed curvature of the lower part of the meniscus may be used as a test. Theory shows that there should be no sensible departure from straightness over a length of about 1 cm.
The Narrow Tube.
For the surface of liquid standing in . a vertical tube of circular section, we have
.                             1        j                  /ix
X Sin ilr = -TH - >- "; ..... 7   xTT = -T,         ZM CtX,     ............... (1 )
Y                                                 a        *
in which z is the vertical co-ordinate measured upwards from the free plane level, x is the horizontal co-ordinate measured from the axis, ty is the angle the tangent at any point makes .with the horizontal, and a? Tgp^, where 27is the surface-tension, g the acceleration of gravity, and p the density of the fluid. The equation expresses the equilibrium of the cylinder of liquid of radius as. At the wall,, where x = r, ty assumes a given value ($TT~ i), and (1) becomes
azr cos i I   zxdx ............................ (2)
If the radius (r) of the tube is small, the total curvature is nearly constant, that is, the surface is nearly spherical    We take
z = 2-V(ca-2) + w,   ...... '.....; ............... (3)
where I is the height of the centre and c the radius of the sphere, while u represents the correction required for a closer approximation.    If we omit u ' altogether, (2) gives
a2 r co s i = \l r* 4- 1 { ( c2 - rrf - c8 } ................... (4)
* Compare Phil. Mag. Vol. xxxiv. p. 309, Appendix, 1892; Scientific Papers, Vol. iv. p. 13.
f The reference is given below.
J It may be remarked that a2 is sometimes taken to denote the double of the above quantity.en that my conclusions differ a good deal from those of Prof. Titchener, but since these estimates depend upon individual judgment, perhaps