An introduction to the mathematical theory of attraction ...

8 Resultant Force.
In this case, by Art. 10, the attraction, at any point P, of
an infinitely long cylinder along whose axis the density is p,
and whose section E is infinitely small, is -2, where p is the
perpendicular drawn from P to the axis of the cylinder, and,
the foot of this perpendicular being Q, the direction of the
attraction is the line PQ. If we draw a plane through P
perpendicular to the generators of the cylinders, it contains
the feet of all the perpendiculars on their axes; and if dS
denote the element of this plane which is enclosed by the
cylinder whose section is e, and r the distance from P to Q,
we have E = dS, and p = r, whence the force at P due to the
-2p dS
thin cylinder is 2pd
r
Hence the resultant force at P is that due to a uniplanar
distribution of mass in the plane through P perpendicular to
the generators of the cylinders, the density r of this distribution at a point Q being 2p, and the force at P duee to any
element of mass varying inversely as its distance.
The student must not confound a uniplanar distribution
of mass, such as has been described above, with a surface
distribution in which the surface happens to be a plane. The
definition of the uniplanar density r is verbally the same as
that of the surface density a, but the two magnitudes are of
different kinds, since in one case the force caused by an
element of mass varies inversely as its distance, and in the
other case inversely as the square of its distance. In fact r
is a magnitude of the same kind as p, the volume density.
The conditions required for a cylindrical distribution of
mass may be approximately fulfilled in nature, and a uniplanar distribution is merely a mathematical artifice by which
the problems belonging to a cylindrical distribution may be
presented in a simpler form.
12. Attraction of Spherical Shell.-To find the
attraction of an infinitely thin homogeneous spherical shell
at an external point P we may proceed as follows:Let O be the centre of the sphere bounding the shell,
draw a plane perpendicular to PO cutting the sphere in a
circle, let Q be any point on this circle, then, if a be the radius