K = 2πσ² ∫∞0 ψ(z) dz. (28)
In Laplace’s investigation σ is supposed to be unity. We may call the value which (28) then assumes K0, so that as above
K0 = 2π ∫∞0 ψ(z) dz. (29)
The expression for the superficial tension is most readily found with the aid of the idea of superficial energy, introduced into the subject by Gauss. Since the tension is constant, the work that must be done to extend the surface by one unit of area measures the tension, and the work required for the generation of any surface is the product of the tension and the area. From this consideration we may derive Laplace’s expression, as has been done by Dupre (Théorie mécanique de la chaleur, Paris, 1869), and Kelvin (“Capillary Attraction,” Proc. Roy. Inst., January 1886. Reprinted, Popular Lectures and Addresses, 1889). For imagine a small cavity to be formed in the interior of the mass and to be gradually expanded in such a shape that the walls consist almost entirely of two parallel planes. The distance between the planes is supposed to be very small compared with their ultimate diameters, but at the same time large enough to exceed the range of the attractive forces. The work required to produce this crevasse is twice the product of the tension and the area of one of the faces. If we now suppose the crevasse produced by direct separation of its walls, the work necessary must be the same as before, the initial and final configurations being identical; and we recognize that the tension may be measured by half the work that must be done per unit of area against the mutual attraction in order to separate the two portions which lie upon opposite sides of an ideal plane to a distance from one another which is outside the range of the forces. It only remains to calculate this work.
If σ1, σ2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area
2πσ1σ2 ∫∞0 ψ(z) dz, (30)
or 2πσ1σ2θ(z), if we write
∫∞0 ψ(z) dz = θ(z) (31)
The work required to produce the separation in question is thus
2πσ1σ2 ∫∞0 θ(z) dz; (32)