Therefore, directions of fields [??], [??] and [??] are a triad of conjugate directions with respect to tensor [mathematical expression not reproducible] and constitutes a local basis, whose reciprocal one is formed by directions of [??], [??] fields and [nabla]S.
For instance, the orthogonality of vectors [[bar.e].sup.1] and [[bar.e].sub.3] (and from Cauchy's theorem , [[bar.e].sup.3] and [[bar.e].sub.1]) implies that [[bar.e].sup.1] and [[bar.e].sup.3] are conjugate directions. In order to visualize these concepts, a section of Cauchy's quadric of revolution is shown (see Fig.
It must be noted that principal directions of a tensor are also conjugate directions. Conjugation arises as a broader concept than that of principal directions (a particular case, where [[bar.[GAMMA]].sub.v] is collinear with V), but keeping the algebraic property of diagonalisation of tensors, if expressed in covariant components.
We have shown that directions of fields [bar.D], [bar.B] and [bar.[upsilon]] (ray velocity) are a triad of conjugate directions with respect to the inverse relative dielectric permittivity tensor and constitutes a local basis, whose reciprocal one is formed by directions of [bar.E], [bar.H] fields and wavevector [bar.k].