M = Sσ, (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 T dS, and the energy of the film will be increased by this amount. Hence
TdS = dE = Sde + edS. (13)
But since M is constant,
dM = Sdσ + σdS = 0. (14)
Eliminating dS from equations (13) and (14), and dividing by S, we find
| T = e − σ | de | . |
| dσ |
In this expression σ 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 ρ is the density, and χ the energy of unit of mass at depth z, then