The exact solutions for the fields inside and outside a dielectric ring in a uniform, axial, electrostatic field are derived in the system of toroidal coordinates. Comparison is made with the perturbation solution generated by inverse aspect ratio expansion, and it is shown how the exact solution may be truncated to any desired accuracy. The cylindrical limits of the exact and truncated solutions are obtained.

1.
W. M.
Hicks
,
Phil. Trans.
176
,
161
(
1884
).
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2.
E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, New York, 1955), p. 433.
3.
Higher Transcendental Functions edited by A. Erdelyi (McGraw‐Hill, New York, 1953), Vol. I.
4.
L. M. Milne Thomsom, The Calculus of Finite Differences (Mac‐Millan, London 1960), p. 531.
5.
A proof of this result is given in Ref. 4, pp. 532–34.
6.
The values of Pn0 and Qn0 are taken from Tables of Associated Legendre Functions, National Bureau of Standards (Columbia U.P., New York, 1945).
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© 1972 The American Institute of Physics.
1972
The American Institute of Physics
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