An introduction to the mathematical theory of attraction ...

Surface Distribution Equivalent to Solid Ellipsoid. 149
CHAPTER V.
SURFACES AND CURVES OF THE SECOND DEGREE.
82. Introductory.-The attraction of a homogeneous
ellipsoid at a point on its surface was investigated in Arts.
21, 22; and in Art. 75, Ex. 5, it was shown that on the
result of this investigation could be based a method of finding
the attraction of an ellipsoid in external space.
This problem is one of great celebrity in the history of
Mathematics, and has been solved by various methods, of
which the most celebrated are those of Maljîlaurin, Chasles,
Ivory, and Thomson. I
A number of expressions for the potential of an ellipsoid
have been given by mathematicians of eminence; and in
consequence of its connexion with the theory of columnar
vortices in a perfect liquid, the determination of the uniplanar potential of a homogeneous elliptic plate is a question
of much interest.
The distribution of electricity on conductors whose surfaces are hyperboloids or paraboloids has been treated by
Maxwell, following Lamé, by means of elliptic coordinates.
It is proposed in the present chapter to give some account
of the results enumerated above.
83. Surface Distribution Equivalent to Solid
Ellipsoid.-If we assume a distribution of mass such that
the potential V is zero at every point outside the surface of
the ellipsoid whose equation is
X2 y2 Z2
2 b2 c+2 -
and that
r- ( eX p a2 Y2 2)
at every point inside this surface, where K is a constant; then