  Central configurations give rise to self-similar solutions to the Newtonian
$N$-body problem, and play important roles in understanding its complicated
dynamics. Even the simple question of whether or not there are finitely many
planar central configurations for $N$ positive masses remains unsolved in most
cases. Considering central configurations as critical points of a function $f$,
we explicity compute the eigenvalues of the Hessian of $f$ for all $N$ for the
point vortex potential for the regular polygon with equal masses. For
homogeneous potentials including the Newtonian case we compute bounds on the
eigenvalues for the regular polygon with equal masses, and give estimates on
where bifurcations occur. These eigenvalue computations imply results on the
Morse indices of $f$ for the regular polygon. Explicit formulae for the
eigenvalues of the Hessian are also given for all central configurations of the
equal mass 4-body problem with a homogeneous potential. Classic results on
collinear central configurations are also generalized to the homogeneous
potential case. Numerical results, conjectures, and suggestions for future work
in the context of a homogeneous potential are given.
