es diffusion takes place
to a limited extent, after which the resulting mixtures do not mix with
each other. The same substance may be able to exist in two different
states at the same temperature and pressure, as when water and its
saturated vapour are contained in the same vessel. The conditions under
which the thermal and mechanical equilibrium of two fluids, two
mixtures, or the same substance in two physical states in contact with
each other, is possible belong to thermodynamics. All that we have to
observe at present is that, in the cases in which the fluids do not mix
of themselves, the potential energy of the system must be greater when
the fluids are mixed than when they are separate.

It is found by experiment that it is only very close to the bounding
surface of a liquid that the forces arising from the mutual action of
its parts have any resultant effect on one of its particles. The
experiments of Quincke and others seem to show that the extreme range of
the forces which produce capillary action lies between a thousandth and
a twenty-thousandth part of a millimetre.

We shall use the symbol [epsilon] to denote this extreme range, beyond
which the action of these forces may be regarded as insensible. If [chi]
denotes the potential energy of unit of mass of the substance, we may
treat [chi] as sensibly constant except within a distance [epsilon] of
the bounding surface of the fluid. In the interior of the fluid it has
the uniform value [chi]0. In like manner the density, [rho], is sensibly
equal to the constant quantity [rho]0, which is its value in the
interior of the liquid, except within a distance [epsilon] of the
bounding surface. Hence if V is the volume of a mass M of liquid bounded
by a surface whose area is S, the integral
_ _ _
/ / /
M = | | | [rho] dx dy dz, (1)
_/_/_/

where the integration is to be extended throughout the volume V, may be
divided into two parts by considering separately the thin shell or skin
extending from the outer surface to a depth [epsilon], within which the
density and other properties of the liquid vary with the depth, and the
interior portion of the liquid within which its properties are constant.

Since [epsilon] is a line of insensible magnitude compared with the
dimensions of the mass of liquid and the principal radii of curvature of
its surface, the volume of the shell whose surface is S and thickness
[epsilon] will be S[epsilon], and