On the Maxwell Representation of the Gravitational Potential for a Symmetric Body

The article analyzes an approach that goes back to Maxwell to the representation of a potential, in particular, the potential of the Newtonian field of gravity as a sum of potentials of multipoles of different orders. Critical cases of the algorithm for finding the parameters of a multipole, namely, its axes and moment, are indicated. The cases take place when the body has certain symmetries in the mass distribution. Recommendations for overcoming the identified difficulties are formulated. For a body with a triaxial ellipsoid of inertia, explicit expressions for the axes and moment of a second-order multipole that are expressed via second-order inertia integrals are given. It is shown that the axes of the multipole are orthogonal to the circular cross-sections of the ellipsoid of inertia of the body. Critical cases of calculating a third-order multipole are considered using the example of a model body with constant density, that has the shape of an equihedral tetrahedron. A method for calculating the axes and moment of a third-order multipole for such a body is given.