An introduction to the mathematical theory of attraction ...

Continuous Distribution of Mass. 5
because it is usually more convenient to count distances from
the acting mass rather than towards it, the standard positive
force when the signs of algebraical expressions have to be
taken into account will be supposed repulsive. When the
forces under consideration are gravitational we may suppose
that the action is between electric masses, one positive and
the other negative, whose numerical expressions are the same
as those for the gravitating masses.
8. Continuous Distribution of 1Iass.-When the
acting mass is continuously distributed, the distribution may
exist throughout a volume, or over a surface, or along a line.
Corresponding to a volume or space distribution, a surface
distribution, and a line distribution, there are three kinds of
density, a volume density denoted by p, a surface density by
o, and a line density by X.
These three magnitudes may be defined as follows:The volume density at a _point Q is the limit of the ratio of
the mass contained by a sphere having Q as centre to the volume
of the sphere 'when its radius is diminished without limit.
The surface density at a point Q, on a surface S, is the limit
of the ratio of the mass contained by a sphere having Q as
centre to the area of the portion of the surface S within the
sphere when its radius is diminished without limit.
The line density at a point Q, on a line s, is the limit of the
ratio of the mass on a portion of s having Q for its middle
point to the length of this portion when it is diminished
without limit.
If p be finite, the corresponding value of a is an infinitely
small quantity of the first order, and that of X an infinitely
small quantity of the second order.
In the case of gravitational forces actually existing in
nature, p is always finite.
For electric forces, on the other hand, c is usually finite.
In both cases we may, for mathematical purposes, suppose
a fictitious surface distribution for which o is finite.
When a is finite, there may, of course, be an independent
volume distribution for which p is finite.
In the case of forces actually existing in nature, X can
never be finite (Clerk Maxwell, "Electricity and Magnetism," Art. 81).