On Maxwell’s Representation of MacCullagh’s Formula: A Way to Determine the Principal Axes of Inertia for a Rigid Body via its Multipole of the Second Order

The well-known MacCullagh formula that considers deviations of a body’s shape from a spherical one for the gravitational potential U of any body at a large distance r from its center of mass to an external test point is presented via Maxwell’s representation for homogeneous harmonic functions in the form of a superposition of directional derivatives of the fundamental solution r⁻¹ of the Laplace equation ∆U = 0. In the case of the general mass distribution, this representation is determined by one scalar value and two unit vectors, h₁ and h₂, located in a plane orthogonal to the middle principal axis of inertia of the body. At the same time, the axis of inertia of the body corresponding to its smallest moment of inertia is the bisector of the angle formed by these vectors. The geometric meaning of the vectors is established: they are orthogonal to the circular sections of the ellipsoid of inertia of the body constructed at its center of mass. This research allows one to propose an approach to finding the central principle axes of inertia of a body according to Maxwell’s representation of its gravitational potential.