  The inversion in the sphere or Kelvin transformation, which exchanges the
radial coordinate for its inverse, is used as a guide to relate distinct
electrostatic problems with dual features. The exact solution of some
nontrivial problems are obtained through the mapping from simple highly
symmetric systems. In particular, the concept of multipole expansion is
revisited from a point of view opposed to the usual one: the sources are
distributed in a region far from the origin while the electrostatic potential
is described at points close to it.
