  Laboratory experiments on gravitation are usually performed with objects of
constant density, so that the analysis of the forces concerns only the geometry
of their shape. In an ideal experiment, the shapes of the constituent parts
will be optimised to meet certain mathematical criteria, which ensure that the
experiment has maximal sensitivity.
  Using this idea, the author suggested an experiment to determine the
departure of the gravitational force from Newton's force law [1]. The
geometrical problem which has to be solved is to find two shapes which differ
significantly, but have the same Newtonian potential. Essentially, the
experiment determines whether the two objects are distinguishable by their
gravitational force. Here, we consider the case when one of them is a round
ball. The result, Theorem 1, establishes a fact which appeared in numerical
simulations, that the second object has to have a hole in it.
