An introduction to the mathematical theory of attraction ...

90 The Potential.
The truth of the equations above is obvious from the
definition of the potential.
52. Change of Energy due to alteration of Mass.
-If the geometrical form of a system remain invariable, but
the mass at each point be altered, the potential receives a
corresponding change, and also the total potential energy
due to the mutual action of the parts of the system on each
other. If the mass at any point be changed from m to m',
the potential at the same point from V to V', and the total
energy of the system from W to W', we have
2 (W'- W) = mn'V'- mV.
If now we suppose the two systems in Art 51 to be geometrically coincident, by (22) we have
zm V' = m' V;
whence
m'V'- mV (= (' -m) (V'+ V)
= (m'+mn) ( '- V);
and we get
W' -w 2Y (qn'- )(/'+ V) = ( + - V F'+ - ). (23)
This equation can be established also in a manner similar
to that employed in Article 50.
EXAMPLES.
1. Show that the component parallel to the axis of x of the repulsion of a
homogeneous body, of density p, bounded by the surface S, at any external
point 0, is equal to the potential of a fictitious distribution on S whose density
at any point of the surface is lp, where l is thé cosine of the angle which the
normal, drawn outwards at the point, makes with the axis of x.
Tale 0 for origin, let X be the force component due to the repelling mass,
and v the potential at 0 of the fictitious surface distribution; then
dxd dz x _ Ix ddy dz dr _ dydz p
X p I Id J dz pl d8
A surface distribution of density lp is merely a mathematical artifice, and
physically impossible (see Art. 8).