An introduction to the mathematical theory of attraction ...

6 Resultant Force.
It is plain that m, À, a, and p are quantities of different
kinds, each being one space-dimension lower than the preceding.
If ds denote a line element, dS a surface element, and dZ
a volume element, the elements of mass corresponding respectively to the three kinds of distribution are given by the
equations
dm = ds, lm = adS, dm = pd~.
Introducing the expression for dm into equations (1) we
have, in the case of a volume distribution,
x=['p ~aY-"dr, z d\. (2)
Similar results may be obtained in like manner for a
surface distribution and a line distribution.
9. Attraction of Thin Cone at its Vertex.-If dw
be the solid angle of the cone, and r the distance from
the vertex of any point in its mass, the corresponding
element of volume is r2 dr do, and the attracting force of the
r2 dr dh
element is p - -, that is pdr d. In this case the elementary forces of attraction are all in the same direction,
and, if the cone be homogeneous, the attraction at the vertex?r+î
of a frustum of length l is pdcw dr or pldw. This is independent of the distance of the frustum from the vertex.
If the frustum extend at both sides of the vertex, the force
exerted by the portion on one side is opposite in direction to
that due to the other portion, and the resultant force is
p (1 - ') dC( towards the portion whose length is 1.
It is plain that this result holds good for any two portions
of the cone which are on opposite sides of the vertex and
whose lengths are I and '.
10. Attraction of Homogeneous Straight IAne.The attraction of a homogeneous straight line AB at any
point P is the same as that of a circular arc of equal density
having P for centre, and the perpendicular distance of P
from AB as radius, the extremities of the arc being on the
lines PA and PB.