Expansions of the negative powers of mutual distances between bodies

The negative powers of the mutual distance^– between two bodies are developed into series converging at any moment but that of collision. On the base of these expansions the series have been constructed representing^– in the perturbation theory of celestial mechanics. In the general case, including intersecting orbits, the terms are quasi-periodic functions of the time. In the case of non-intersecting orbits the expansion is a double Fourier series in the mean anomalies. All the expansions have a literal form with respect to osculating elements.