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and then (5.15b) requires that the constant of integration be zero. The condition
(5.15c) now shows that

which defines the height of the column of the liquid (measured at the centre-line of
the tube). The solution for can be written down immediately; it is conveniently
expressed in terms of a parameter as

which satisfies condition (5.15a). We have found the first approximation to the height
of the liquid at r = 0 (namely, and the shape of the surface of the liquid
inside the tube: a section of a spherical shell. Further terms in the asymptotic expansions
can be found quite routinely; these expansions are uniformly valid for

Exterior problem
Finding the solution for the shape of the surface outside the tube is technically a more
demanding exercise. First, the deviation of the surface from its level (z = 0) at infinity
is observed to be not particularly large—as compared with what happens inside. This
suggests that we attempt to solve equation (5.13) directly, subject to

Let us write

so that we are not committing ourselves, at this stage, to the size of the next term in
the asymptotic expansion; in fact, as we shall see, logarithmic terms arise, although we
will not pursue the details here. The equation for is simply

where the arbitrary constant A is determined from (5.16a) as One further
integration of (5.17) then yields