Mutual energy of Gaussian rings

A problem of mutual potential energy of two elliptical gravitating (or electrostatically charged) Gaussian rings is formulated and solved. The rings are coplanar, and their apsid lines generally have an angle of inclination to each other. The mutual energy of the rings is found in quadratic approximation with respect to ring eccentricities $e_1$ and $e_2$. At the first stage, the potential of the Gaussian ring is represented as a series in terms of eccentricity and determined at the points of another elliptical ring (note theoretical importance of such a result). Linear (with respect to quantities $e_{1}$ and $e_{2}$) terms are absent in the expression for the mutual energy of the rings, and the coefficients of the second-order terms ($e_1^2$ and $e_2^2$) are equal to each other. Only one coefficient of mixed term ($e_{1}e_{2}$) is determined by the tilt angle of the apse lines. Such a result can be used to easily determine the moment of force between the rings that is needed for the study of small mutual oscillations of the Gaussian rings.