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 mm' Cf^-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
[phi](f), where [phi](f) is a function of f which is insensible for
all sensible values of f, but which becomes sensible and even
enormously great when f is exceedingly small.
If we next introduce a new function of f and write
_
/ [oo]
| [phi](f)df = [Pi](f), (23)
_/f
then mm' [Pi](f) will represent--(1) The work done by the attractive
force on the particle m, while it is brought from an infinite distance
from m' to the distance f 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 [Pi](f) is also insensible for sensible
values of f, but for insensible values of f it may become sensible and
even very great.
If we next write
_
/ [oo]
| f[Pi](f)df = [psi](z), (24)
_/z
then 2[pi]m[sigma][psi](z) will represent--(1) The work done by the
attractive force while a particle m is brought from an infinite
distance to a distance z from an infinitely thin stratum of the
substance whose mass per unit of area is [sigma]; (2) The attraction
of a particle m placed at a distance z from the plane surface of an
infinite solid whose density is [sigma].
[Illustration: FIG. 2]
Let us examine the case in which the particle m is placed at a
distance z from a curved stratum of the substance, whose principal
radii of curvature are R1 and R2. Let P (fig. 2) be the particle and
PB a normal to the surface. Let the plane of the paper be a normal
section of the surface of the stratum at the point B, making an angle
[omega] with the section whose radius of curvature is R1. Then if O is
the centre of curvature in the plane of the paper, and BO = u,
1 cos^2[omega] sin^2[omega]
-- = ------------ + -----------. (25)
u R1 R2
Let POQ = [theta], PO = r, PQ = f, BP = z,
f^2 = u^2 + r^2 - 2ur cos[theta]. (26)