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 *y*-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) | |||

(12) |

where , , and . In these coordinates,

(13) | |||

(14) | |||

(15) |

(Abramowitz and Stegun 1972). The Scale Factors are

(16) | |||

(17) | |||

(18) |

and the Laplacian is

(19) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Definition of Oblate 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 (, , ).'' §2.11 in
*Mathematical Methods for Physicists, 2nd ed.* Orlando, FL: Academic Press, pp. 107-109, 1970.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.* New York: McGraw-Hill, p. 663, 1953.

© 1996-9

1999-05-26