A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the
Elliptic 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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1999-05-26