The common conjugate directions of plane sections of two concentric ellipsoids

Abstract

This paper is concerned with the analytical solution of two related mathematical problems which arise in mechanics and optics. Problem I consists of determining analytically for a given ellipsoid the directions of the principal axes of any plane elliptical section. It arises when dealing with electromagnetic plane waves in homogeneous, non-conducting, non-opti-cally active, magnetically isotropic media, characterized by an electric permittivity tensor k (see, for instance [1] or [2]). Indeed, the directions of vibration of the electric displacement D of the two homogeneous waves that may propagate in any direction n lie along the principal axes of the elliptical section by the plane n · x = 0 of the “index ellipsoid” x · k ^-1 x = 1.