Poincaré Second Species Solutions of the General Three-Body Problem

We consider the plane three-body problem with two of the masses much smaller than the third. Periodic solutions with near collisions of the small bodies were named by Poincaré second species solutions. Poincaré gave an idea of how to prove their existence, but didn’t provide an actual proof. Complete proofs appeared much later and only for the restricted three-body problem. We reduce the problem of second species solutions of the general threebody problem to the study of dynamics of a billiard type problem and show how to transform the arguments of Poincaré into a rigorous proof.