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A system of curvilinear coordinates variously denoted 
(Arfken 1970) or 
(Moon and Spencer 1988). Using the notation
of Arfken, the bispherical coordinates are defined by
Surfaces of constant 
are given by the spheres
 |
(4) |
surfaces of constant 
by apple surfaces (
)
or lemon surfaces (
)
 |
(5) |
and surface of constant 
by the half-planes
 |
(6) |
The scale factors are
The Laplacian is given by
 |
(10) |
In bispherical coordinates, Laplace's equation is separable (Moon and Spencer 1988), but the Helmholtz differential equation is not.
See also
Bicyclide Coordinates, Laplace's Equation--Bispherical Coordinates, Spherical Coordinates, Toroidal Coordinates
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References
Arfken, G. "Bispherical Coordinates 
." ยง2.14 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117,
1970.Moon, P. and Spencer, D. E. "Bispherical Coordinates
."
Fig. 4.03 in Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, pp. 110-112, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666,
1953.
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Cite this as:
Weisstein, Eric W. "Bispherical Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BisphericalCoordinates.html