Planetary theories with the aid of the expansions of elliptic functions

For coplanar circular orbits, the mutual perturbations between two bodies can be expressed in term of the argument of Jacobian elliptic functions instead of the difference of the mean longitudes. For a given pair of planets, such a change of time variable improves the convergence of the developments. At the first order of planetary masses an integration of Lagrange's equations for the osculating elements is performed. When compared to classical developments the results are reduced by an important factor. The method is then extended to the mutual perturbations of Jupiter and Saturn, at any order of planetary masses, either with Fourier series with two arguments, or with one argument solely, taking advantage of the close commensurability of the mean motions.