On the reducibility of the ellipsoidal system

Lame introduced a triaxial ellipsoidal system, and via some ingenious arguments, he managed to spectrally decompose the Laplace operator and define ellipsoidal harmonic functions. Since then, many authors modified the Lame system and proposed some related ellipsoidal coordinate variables. Perhaps the most important of them is the system introduced by Jacobi, in connection with the geodesic curves on an ellipsoid. All other attempts to introduce ellipsoidal coordinates are basically modifications of the Jacobi or the Lame system. However, an important question in the theory of ellipsoidal harmonics is the way in which this theory reduces to the spheroidal and spherical systems. This is by no means a straightforward procedure, since the relative limiting cases lead to generic underdetermined forms. The basic difficulty is due to the different dimensionality of the singularity regions corresponding to each system. Note that this region has zero dimensions in the case of the sphere, one dimension in the case of the prolate spheroid, and two dimensions in the case of the oblate spheroid and the ellipsoid. The present work provides a systematic way to obtain these reductions and to establish a correspondence between ellipsoidal, spheroidal, and spherical harmonics. We have also demonstrated how our approach applies to the proposed modified Jacobi ellipsoidal system when we connect the Lame with the Jacobi system through certain non-degenerable relations. The importance of these geometrical degeneracies lies in the fact that the solution of any boundary value problem in ellipsoidal geometry provides immediately the solutions in all special geometries of prolate and oblate spheroids, discs, needles, or spheres.