An introduction to the mathematical theory of attraction ...

68 The Potential.
the radius of the sphere, and 0 the angle which the radius
drawn to any point of its surface makes with CP; taking
P for origin, we have
I a sin q~ do 27rra a dr
V = 27= -- 1dr,
r c
since r2= a2 + 2 - 2ac cos 9.
If P be outside the sphere, the limits of the integral are
c + a and c - a; if P be inside, they are a + c and a - c. Hence
for an external point V= and for an internal F= 4wraa.
If now we suppose the point P variable, and take C for
origin, we have
4A7ra2
V= -, or V= 47raa,
according as P is external or internal.
If P be on the surface of the sphere, the value of V for
one form of the function is the same as for the other.
From the equations for V given above, we learn that the
potential of a homogeneous spherical shell has the same
constant value for all points inside it; and for all external
points is the same as if the entire mass of the shell were
concentrated at its centre.
From this last result we may conclude that the potential
of a homogeneous solid sphere at a point P outside it is given
by the equation V = -, where M is the mass of the sphere,
and r the distance of P from its centre. The same equation
holds good if the sphere be composed of homogeneous layers,
comprised between spheres concentric with the external
surface.
The potential of a homogeneous solid sphere at an internal
point P, whose distance from the centre C is r, is the sum of
the potential of the concentric sphere passing through P
and of the thick shell comprised between this sphere and
the external boundary. If the radius of this latter be a,