When a liquid is in thermal and dynamical equilibrium with its vapour, then if ρ′ and χ′ are the values of ρ and χ for the vapour, and ρ0 and χ0 those for the liquid,
χ′ − χ0 = JL − p(1/ρ′ − 1/ρ0), (21)
where J is the dynamical equivalent of heat, L is the latent heat of unit of mass of the vapour, and p is the pressure. At points in the liquid very near its surface it is probable that χ is greater than χ0, and at points in the gas very near the surface of the liquid it is probable that χ is less than χ′, but this has not as yet been ascertained experimentally. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics.
We have next to determine the value of χ in terms of the action between one particle and another. Let us suppose that the force between two particles m and m’ at the distance f is
F = mm′ (φ(ƒ) + Cƒ-2), (22)
being reckoned positive when the force is attractive. The actual force between the particles arises in part from their mutual gravitation, which is inversely as the square of the distance. This force is expressed by m m′ Cƒ-2. It is easy to show that a force subject to this law would not account for capillary action. We shall, therefore, in what follows, consider only that part of the force which depends on φ(ƒ), where φ(ƒ) is a function of ƒ which is insensible for all sensible values of ƒ, but which becomes sensible and even enormously great when ƒ is exceedingly small.
If we next introduce a new function of f and write
∫∞ƒ φ(ƒ) dƒ = Π (ƒ), (23)
then m m′ Π(ƒ) will represent—(I) The work done by the attractive force on the particle m, while it is brought from an infinite distance from m′ to the distance ƒ from m′; or (2) The attraction of a particle m on a narrow straight rod resolved in the direction of the length of the rod, one extremity of the rod being at a distance f from m, and the other at an infinite distance, the mass of unit of length of the rod being m′. The function Π(ƒ) is also insensible for sensible values of ƒ, but for insensible values of ƒ it may become sensible and even very great.
If we next write