On some orthogonality properties of Maxwell's multipole vectors

We determine the location of the expansion points with respect to which the two Maxwell's multipole vectors of the quadrupole moment and the dipole vector of a distribution of charge form an orthogonal trihedron. We find that with respect to these 'orthogonality centres' both the dipole and the quadrupole moments are each characterized by a single real parameter. We further show that the orthogonality centres coincide with the stationary points of the magnitude of the quadrupole moment and, therefore, they can be seen as an extension of the concept of centre of the dipole moment of a neutral system introduced previously in the literature. The nature of the stationary points then provides the means for the classification of a distribution of charge in two different categories.