  We consider the $n$ body problem defined on surfaces of constant positive
curvature. For the 5 and 7 body problem in a collinear symmetric configuration
we obtain initial positions which lead to relative equilibria. We give
explicitly the values of masses in terms of the initial positions. For
positions for which relative equilibria exist, there are infinitely many values
of the masses that generate such solutions. For the 5 and 7 body problem, the
set of parameters (masses and positions) leading to relative equilibria has
positive Lebesgue measure.
