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XII.—Paraboloidal Co-ordinates and Laplace's Equation*†
Published online by Cambridge University Press: 14 February 2012
Synopsis
In this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 66 , Issue 3 , 1963 , pp. 129 - 139
- Copyright
- Copyright © Royal Society of Edinburgh 1963
References
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