  We consider the motion of point masses given by a natural extension of
Newtonian gravitation to spaces of constant positive curvature. Our goal is to
explore the spectral stability of tetrahedral orbits of the corresponding
4-body problem in the 2-dimensional case, a situation that can be reduced to
studying the motion of the bodies on the unit sphere. We first perform some
extensive and highly precise numerical experiments to find the likely regions
of stability and instability, relative to the values of the masses and to the
latitude of the position of three equal masses. Then we support the numerical
evidence with rigorous analytic proofs in the vicinity of some limit cases in
which certain masses are either very large or negligible, or the latitude is
close to zero.
