  Static solutions of the electro-gravitational field equations exhibiting a
functional relationship between the electric and gravitational potentials are
studied. General results for these metrics are presented which extend previous
work of Majumdar. In particular, it is shown that for any solution of the field
equations exhibiting such a Weyl-type relationship, there exists a relationship
between the matter density, the electric field density and the charge density.
It is also found that the Majumdar condition can hold for a bounded perfect
fluid only if the matter pressure vanishes (that is, charged dust). By
restricting to spherically symmetric distributions of charged matter, a number
of exact solutions are presented in closed form which generalise the
Schwarzschild interior solution. Some of these solutions exhibit functional
relations between the electric and gravitational potentials different to the
quadratic one of Weyl. All the non-dust solutions are well-behaved and, by
matching them to the Reissner-Nordstr\"{o}m solution, all of the constants of
integration are identified in terms of the total mass, total charge and radius
of the source. This is done in detail for a number of specific examples. These
are also shown to satisfy the weak and strong energy conditions and many other
regularity and energy conditions that may be required of any physically
reasonable matter distribution.
