s mass, and E its energy; [sigma]
the mass, and e the energy of unit of area; then

M = S[sigma], (11)

E = Se. (12)

Let us now suppose that by some change in the form of the boundary of
the film its area is changed from S to S + dS. If its tension is T the
work required to effect this increase of surface will be TdS, and the
energy of the film will be increased by this amount. Hence

TdS = dE = Sde + edS. (13)

But since M is constant,

dM = Sd[sigma] + [sigma]dS = 0. (14)

Eliminating dS from equations (13) and (14), and dividing by S, we
find

de
T = e - [sigma]--------, (15)
d[sigma]

In this expression [sigma] denotes the mass of unit of area of the
film, and e the energy of unit of area.

If we take the axis of z normal to either surface of the film, the
radius of curvature of which we suppose to be very great compared with
its thickness c, and if [rho] is the density, and [chi] the energy of
unit of mass at depth z, then
_
/ c
[sigma] = | [rho] dz, (16)
_/0

and
_
/ c
e = | [chi] [rho] dz. (17)
_/0

Both [rho] and [chi] are functions of z, the value of which remains
the same when z - c is substituted for z. If the thickness of the film
is greater than 2 [epsilon], there will be a stratum of thickness c -
2 [epsilon] in the middle of the film, within which the values of
[rho] and [chi] will be [rho]0 and [chi]0. In the two strata on either
side of this the law, according to which [rho] and [chi] depend on the
depth, will be the same as in a liquid mass of large dimensions. Hence
in this case
_
/ [epsilon]
[sigma] = (c - 2[epsilon]) [rho]0 + 2 | [rho]d[nu], (18)
_/0
_
/ [epsilon]
e = (c - 2[epsilon]) [chi]0[rho]0 + 2 | [chi][rho]d[nu], (19)
_/ 0

d[sigma] de de
-------- = [rho]0, -- = [chi]0[rho]0, .: -------- = [chi]0,
dc dc d[sigma]
_ _
/ [epsilon]