Spheroconal coordinates elliptical cone

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

The construction of the Lam6 spheroconal harmonic polynomials involves Eq. (43) with matching parameters Eq. (44), matching species, and matching excitations of the respective elliptical cone coordinate degrees of freedom. The matching of species and kinds are fhe following ... 

Notice the common radial coordinate and scale factor for both spherical and spheroconal coordinates, Equations (45) and (46). Fixed values of Xi correspond to elliptical cones with a common vertex at the origin, and an axis along the z-axis for i = 1 and along the x-axis for i = 2. 

This type of confinement is alternative to that by a circular cone in [22] and Section 4.2, using spheroconal coordinates instead of spherical coordinates. The radial coordinate is common to both sets, but the angular coordinates are different. Here we first identify the set of cones by fixed values of the coordinate xi with their common vertex as the origin, their common axis as the z-axis, and elliptical cross sections as determined next. In fact, by using the relationships... 

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