Using the Notation of Byerly (1959, pp. 252-253), Laplace's Equation can be reduced to

(1) |

(2) | |||

(3) | |||

(4) |

In terms of , , and ,

(5) | |||

(6) | |||

(7) |

Equation (1) is not separable using a function of the form

(8) |

(9) | |||

(10) | |||

(11) |

These give

(12) | |||

(13) |

and all others terms vanish. Therefore (1) can be broken up into the equations

(14) | |||

(15) | |||

(16) |

For future convenience, now write

(17) | |||

(18) |

then

(19) | |||

(20) | |||

(21) |

Now replace , , and to obtain

(22) | |||

(23) | |||

(24) |

Each of these is a Lamé's Differential Equation, whose solution is called an Ellipsoidal Harmonic. Writing

(25) | |||

(26) | |||

(27) |

gives the solution to (1) as a product of Ellipsoidal Harmonics .

(28) |

**References**

Arfken, G. ``Confocal Ellipsoidal Coordinates
.'' §2.15 in
*Mathematical Methods for Physicists, 2nd ed.* Orlando, FL: Academic Press, pp. 117-118, 1970.

Byerly, W. E. *An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.* New York: Dover, pp. 251-258, 1959.

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

© 1996-9

1999-05-25