  Newton's equations of celestial mechanics are shown to possess a continuum of
solutions in which the future trajectories of the N bodies are a perfect
reflection of their past. These solutions evolve from zero initial velocities
of the N bodies. Consequently, the future gravitational forces acting on the N
bodies are also a perfect reflection of their past. The proof is carried out
via Taylor series expansions. A perturbed system of equations of the N body
problem is also considered. All real valued solutions of this perturbed system
have no singularities on the real line. The perturbed system is shown to have a
continuum of solutions that possess symmetry where the future velocities of the
N bodies are a perfect reflection of their past. The positions and
accelerations of the N bodies are then odd functions of the time. All N bodies
then evolve from one location in space.
