Prolate Spheroidal Coordinates
A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of theElliptic Cylindrical Coordinates about the x-Axis, which is relabeled the z-Axis. The third set of coordinates consists of planes passing through this axis.
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
where
, 
, and 
. Arfken (1970) uses 
instead of
. The Scale Factors are | |  | (4) |
 | | | (5) |
 | |  | (6) |
The Laplacian is
 | |
 | (7) |
 | (8) |
An alternate form useful for ``two-center'' problems is defined by
 | |  | (9) |
 | |  | (10) |
 | |  | (11) |
where
, 
, and 
(Abramowitz and Stegun 1972). In these coordinates, | |  | (12) |
| |  | (13) |
| |  | (14) |
In terms of the distances from the two Foci,
| |  | (15) |
| |  | (16) |
 | |  | (17) |
The Scale Factors are
 | |  | (18) |
 | |  | (19) |
 | |  | (20) |
and the Laplacian is
 | |
 | (21) |
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Prolate Spheroidal Coordinates.'' §21.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 752, 1972.
Arfken, G. ``Prolate Spheroidal Coordinates (
, 
, 
).'' §2.10 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 103-107, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 661, 1953.
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