ma] de
-------- = [rho], -- = [rho]([chi]' - 4[pi][rho][theta](0)) + 4[pi][rho]^2[theta](c).
dc dc

Hence the surface-tension
_
de / / c \
T = e - [sigma] -------- = 4[pi][rho]^2 ( | [theta](z)dz - c[theta](c) ).
d[sigma] \ _/0 /

Integrating the first term within brackets by parts, it becomes
_
/ c d[theta]
c[theta](c) - 0[theta](0) - | z -------- dz.
_/0 dz

Remembering that c(0) is a finite quantity, and that

d[theta]
-------- = -[psi](z),
dz

we find
_
/ c
T = 4[pi][rho]^2 | z[psi](z)dz. (27)
_/0

When c is greater than [epsilon] this is equivalent to 2H in the
equation of Laplace. Hence the tension is the same for all films
thicker than [epsilon], the range of the molecular forces. For thinner
films

dT
-- = 4[pi][rho]^2c[psi](c).
dc

Hence if [psi](c) is positive, the tension and the thickness will
increase together. Now 2[pi]m[rho][psi](c) represents the attraction
between a particle m and the plane surface of an infinite mass of the
liquid, when the distance of the particle outside the surface is c.
Now, the force between the particle and the liquid is certainly, on
the whole, attractive; but if between any two small values of c it
should be repulsive, then for films whose thickness lies between these
values the tension will increase as the thickness diminishes, but for
all other cases the tension will diminish as the thickness diminishes.

We have given several examples in which the density is assumed to be
uniform, because Poisson has asserted that capillary phenomena would
not take place unless the density varied rapidly near the surface. In
this assertion we think he was mathematically wrong, though in his own
hypothesis that the density does actually vary, he was probably right.
In fact, the quantity 4[pi][rho]^2K, which we may call with van der
Waals the molecular pressure, is so great for most liquids (5000
atmospheres for water), that in the parts near the surface, where the
molecular pressure varies rapidly, we may