[sigma]u^2 sin[theta] d[theta] d[omega],
or expressing d[Greek: th] in terms of df by (26),
[sigma]ur^-1 f df d[omega].
Multiplying this by m and by [pi](f), we obtain for the work done by
the attraction of this element when m is brought from an infinite
distance to P1,
m[sigma]ur^-1 f[Pi](f)dfd[omega].
Integrating with respect to f from f = z to f = a, where a is a line
very great compared with the extreme range of the molecular force, but
very small compared with either of the radii of curvature, we obtain
for the work
_
/
| m[sigma]ur^-1 ([psi](z) - [psi](a))d[omega],
_/
and since [psi](a) is an insensible quantity we may omit it. We may
also write
ur^-1 = 1 + zu^-1 + &c.,
since z is very small compared with u, and expressing u in terms of
[omega] by (25), we find
_ _ _
/ 2[pi] | /cos^2[omega] sin^2[omega]\ |
| m[sigma][psi](z) | 1 + z ( ------------ + ------------ ) | d[omega] =
_/0 |_ \ R1 R2 / _|
_ _
| 1 /1 1 \ |
2[pi]m[sigma][psi](z) | 1 + --z ( -- + -- ) |.
|_ 2 \R1 R2/ _|
This then expresses the work done by the attractive forces when a
particle m is brought from an infinite distance to the point P at a
distance z from a stratum whose surface-density is [sigma], and whose
principal radii of curvature are R1 and R2.
To find the work done when m is brought to the point P in the
neighbourhood of a solid body, the density of which is a function of
the depth [nu] below the surface, we have only to write instead of
[sigma][rho]dz, and to integrate
_ _
/ [oo] /1 1 \ / [oo]
2[pi]m | [rho][psi](z)dz + [pi]m ( -- + -- ) | [rho]z[psi](z)dz,
_/z \R1 R2/ _/z
where, in general, we must suppose [rho] a function of z. This
expression, when integrated, gives (1) the work done on a particle m
while it is brought from an infinite distance to the point P, or (2)
the attraction on a long slender column normal to the surface and
terminating at P, the mass of unit of length of the column being m. In
the form of the theory given by Laplace, the density of the liquid was
supposed to be uniform. Hence if we write
_ _
/ [oo] / [oo]
K = 2[pi] | [psi](z)dz, H = 2[pi] | z[psi](z)dz,
_/0 _/0
the pressure of a column _of the fluid itself_ terminating at the
surface will be
[rho]^2 {K + 1/2H(1/R1 + 1/R2)},
and the work done by the attractive forces when a particle m is
brought to the surface of the fluid from an infinite distance will be
m[rho] {K + 1/2H(1/R1 + 1/R2)},
If we write
_
/ oo
| [psi](z)dz = [theta](z),
_/z
then 2[pi]m[rho][theta](z) will express the work done by-the
attractive forces, while a particle m is brought from an infinite
distance to a distance z from the plane surface of a mass of the
substance of density [rho] and infinitely thick. The function
[theta](z) is insensible for all sensible values of z. For insensible
values it may become sensible, but it must remain finite even when z =
0, in which case [theta](0) = K.
If [chi]' is the potential energy of unit of mass of the substance in
vapour, then at a distance z from the plane surface of the liquid
[chi] = [chi]' - 2[pi][rho][theta](z).