  The monopole solution of the Einstein vacuum field equations (Schwarzschild`s
solution) in Weyl coordinates involves a metric function that can be
interpreted as the gravitational potential of a bar of length $2m$ with
constant linear density. The question addressed in this work is whether similar
representations can be constructed for Weyl solutions other than the
spherically symmetric one.
  A new family of static solutions of the axisymmetric vacuum field equations
generalizing the M-Q$^{(1)}$ solution is developed. These represent slight
deviations from spherical symmetry in terms of the relativistic multipole
moments (RMM) we wish the solution to contain. A Newtonian object referred to
as a dumbbell can be used to describe these solutions in a simple form by means
of the density of this object, since the physical properties of the
relativistic solution are characterized by its behaviour. The density profile
of the dumbbell, which is given in terms of the RMM of the solution, allows us
to distinguish general multipole Weyl solutions from the constant-density
Schwarzschild solution. The range of values of the multipole moments that
generate positive-definite density profiles are also calculated. The bounds on
the multipole moments that arise from this density condition are identical to
those required for a well-behaved infinite-redshift surface $g_{00}=0$.
