Geometry of the three-particle problem on the line

A modification of the Delves coordinates is considered for the problem of three particles on a fixed straight line, that interact with each other through the two-particle potentials Vij ∼ rij-1. A number of new geometrical objects related to this problem is introduced: the triangle of positions of pair collisions Δ(P12, P23, P31), geometrical positions of three particles at fixed hyperradius (three circles Ci), etc. The triple and pair collisions in the system, Hill's curves of the zero velocity for the given case and librational directions are described in geometrical terms. The geometrical solution of the proposed equation for librational directions is expounded as well as the solution of this equation by the power series at three typical sets of values of particle masses: three masses comparable in magnetude, two heavy and one light particle and one heavy and two light particles. The role is discussed of the reduced mass of the three-particle system as a source of nonanalytical singularities in the problem independent of those caused by particle collisions