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).

At the surface

[chi] = [chi]' - 2[pi][rho][theta](0).

At a distance z within the surface

[chi] = [chi]' - 4[pi][rho][theta](0) + 2[pi][rho][theta](z).

If the liquid forms a stratum of thickness c, then

[chi] = [chi]' - 4[pi][rho][theta](0) + 2[pi][rho][theta](z) +

+ 2[pi][rho][theta](z - c).

The surface-density of this stratum is [sigma] = c[rho]. The energy
per unit of area is
_
/ c
e = | [chi][rho]dz = c[rho]([chi]' - 4[pi][rho][theta](0)) +
_/0
_ _
/ c / c
+ 2[pi][rho]^2 | [theta](z)dz + 2[pi][rho]^2 | [theta](z - c)dz.
_/0 _/0

Since the two sides of the stratum are similar the last two terms are
equal, and
_
/ c
e = c[rho]([chi]' - 4[pi][rho][theta](0)) + 4[pi][rho]^2 | [theta](z)dz.
_/0

Differentiating with respect to c, we find

d[sig