On the problem of homographic solutions of the three-body problem

Abstract

This paper deals with the problem of the existence of homographic solutions of the three-body problem in the case of the law of the inverse cube of the mutual distances. It is discussed a solution given by Dr. R.P. Cesco in his paper "Sobre las soluciones homográficas del problema de los tres cuerpos", Pub. Observ. Astron. La Plata, Serie Astronómica, Tomo XXV , Nº2,1959. We quote from this paper: "The object of this paper is to prove the following theorem:The only homographic solutions of the three-body problem of celestial mechanics with a law of attraction proportional to any power r α of the distance r are - (I) The pure dilatation. (II) The collinear solutions. (III) The equilateral solutions. (IV) The isosceles solutions of Banachiewitz for α = 3, and (V) The scalene solutions given in this note, also for α=3 ,the first three kinds being the only planar solutions for any value of α".