The coordinates are the asymptotic angle of confocal Parabola segments symmetrical about the axis. The
coordinates are confocal Ellipses centered on the origin.

(1) | |||

(2) | |||

(3) |

where , , and . They are related to Cartesian Coordinates by

(4) |

(5) |

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) |

The Laplacian is

(13) |

(14) | |||

(15) | |||

(16) |

Then the new Scale Factors are

(17) | |||

(18) | |||

(19) |

The Helmholtz Differential Equation is Separable.

**References**

Arfken, G. ``Elliptic Cylindrical Coordinates (, , ).'' §2.7 in
*Mathematical Methods for Physicists, 2nd ed.* Orlando, FL: Academic Press, pp. 95-97, 1970.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.* New York: McGraw-Hill, p. 657, 1953.

© 1996-9

1999-05-25