  This is an investigation into the exact forms of images of point charges in
2D conducting ellipses and 3D prolate spheroids. For an ellipse with exterior
point source, we compare two previous expressions for the analytic continuation
inside the ellipse down to the focal line, the location of the image charge.
For an interior source, we discover a sequence of image charges lying outside.
For a point source close the surface of a spheroid, series solutions for the
potential diverge in a region that encompasses the singularities of the
continuation of the reflected potential inside the spheroid. We uncover an
image system for a point charge on the axis of a prolate spheroid that extends
from the focal segment somewhat further up the axis. For an off axis charge we
find a new approximate image charge. For a point charge at the center of the
spheroid, the image singularity is found to lie on an infinitely wide, flat
sheet with a hole around the spheroid. For a point charge located anywhere
inside the spheroid, we find two new approximate image charge solutions, a
point charge in real space which applies when source lies near the surface, or
a point charge in complex space which projects as a ring in the real space
which applies when the source lies near the rotation axis. When the point
charge lies exactly on a focal point, a multipole of order 1/2 is an ideal
image approximation.
