The hyperellipsoidal system.

One hundred ninety years ago, Lamé introduced the celebrated ellipsoidal system, which is an orthogonal curvilinear system that incorporates any anisotropic characteristics of three‐dimensional space. Since then, the theory of ellipsoidal harmonics has been successfully developed, but almost nothing has been achieved for ellipsoidal waves. This is due to the fact that, in order to deal with the ellipsoidal harmonics, Lamé introduced some ingenious arguments that led to a complete spectral theory for the Kernel space of the Laplacian, but no such arguments have been proposed for the Kernel space of the d'Alembert operator. So, one possible approach to develop a method that will give us ellipsoidal wave functions is to try to generalize the Lamé arguments (to the extent that this is possible) to four‐dimensional ellipsoidal harmonics. This procedure will give us the hyperellipsoidal harmonics. Then we can replace the fourth coordinate by ict in order for the four‐dimensional Laplace operator to become the three‐dimensional wave operator of d'Alembert and the four‐dimensional harmonics to become three dimensional waves. The present work is the first stage of this project. It develops the geometry of a four‐dimensional ellipsoidal system and calculates the Laplace operator in this hyperellipsoidal system. [ABSTRACT FROM AUTHOR]