Among the eleven systems of coordinates [8] that allow a separation of three spatial variables in the Helmholtz equation, or hence also the Laplace equation because that Helmholtz equation contains the laplacian operator, only four systems enable a complete separation of variables of Schroedinger's partial-differential equation for the hydrogen atom to yield ordinarydifferential equations, specifically the two specified above plus

ellipsoidal coordinates [9], for which only indirect solutions in series had been achieved [20] before the present work, and spheroconical coordinates for which no explicit algebraic solution has ever been reported.

The normal gravity potential U(u,[beta]) can be expressed in a form of the

ellipsoidal coordinates u and [beta],

From a chemical or physical point of view, a consideration of the H atom in paraboloidal coordinates seems more important than in spherical polar coordinates, but

ellipsoidal coordinates are more practical than either spherical polar or paraboloidal.

where [phi], [lambda] are

ellipsoidal coordinates of the involved stations.

In this part III of a series of articles devoted to the hydrogen atom with its coordinates separable in these four systems, we state the temporally independent partial-differential equation and its solution in ellipsoidal coordinates, and provide plots of selected amplitude functions as surfaces corresponding to a chosen value of amplitude.

SCHROEDINGER'S TEMPORALLY INDEPENDENT EQUATION IN ELLIPSOIDAL COORDINATES

Deflection components determination consist in the comparison of astronomical coordinates: latitude and longitude and

ellipsoidal coordinates of the same point (Hofmann-Wellenhof and Moritz, 2006).

Although a confirmation that a separation of coordinates in a molecular context is practicable also in

ellipsoidal coordinates in an application to [H.

The information about point's

ellipsoidal coordinates and normal heights makes excellent opportunities for determination of quasigeoid or geoid undulation.

In

ellipsoidal coordinates, one focus of an ellipsoid is located at or near the atomic nucleus; the other focus, at distance d, is merely a dummy location; as the latter can become the location of a second atomic nucleus, the associated amplitude functions become formally applicable to a diatomic molecule, which has been the reason for the attention given to these coordinates [9].

where [phi], [lambda] are

ellipsoidal coordinates of the point with orthogonal coordinates x, y, z and v are observed components of velocities.

For coordinates in two additional systems as variables in which Schroedinger's partial-differential equations are separable [7] and for which we here generate direct algebraic expressions for the first time, the solutions are investigated little and indirectly in

ellipsoidal coordinates [8], and even less, without explicit formulae, in spheroconical coordinates [9].