σ = ∫c0 ρdz, (16)
and
e = ∫c0 χρdz. (17)
Both ρ and χ 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 ε, there will be a stratum of thickness c − 2ε in the middle of the film, within which the values of ρ and χ will be ρ0 and χ0. In the two strata on either side of this the law, according to which ρ and χ depend on the depth, will be the same as in a liquid mass of large dimensions. Hence in this case
σ = (c − 2ε) ρ0 + 2 ∫ε0 ρdν, (18)
e = (c − 2ε) χ0ρ0 + 2 ∫ε0 χρdν, (19)
| dσ | = ρ0, | de | = χ0ρ0, ∴ | de | = χ0, |
| dc | dc | dσ |
T = 2 ∫ε0 χρ dν − 2χ0 ∫ε0 ρdν = 2 ∫ε0 (χ − χ0) ρdν. (20)
Hence the tension of a thick film is equal to the sum of the tensions of its two surfaces as already calculated (equation 7). On the hypothesis of uniform density we shall find that this is true for films whose thickness exceeds ε.
The symbol χ is defined as the energy of unit of mass of the substance. A knowledge of the absolute value of this energy is not required, since in every expression in which it occurs it is under the form χ − χ0, that is to say, the difference between the energy in two different states. The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. It is impossible to make direct measurements of the properties of particles of the substance within the insensible distance ε of the bounding surface.