`Similar' coordinate systems and the Roche geometry. Application

A new equivalence relation, named relation of ‘similarity’ is defined and applied in the restricted three-body problem. Using this relation, a new class of trajectories (named ‘similar’ trajectories) are obtained; they have the theoretical role to give us new details in the restricted three-body problem. The ‘similar’ coordinate systems allow us in addition to obtain a unitary and an elegant demonstration of some analytical relations in the Roche geometry. As an example, some analytical relations published in Astrophysical Journal by Seidov in 2004 are demonstrated.