Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/230

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The quantity of electricity on a planetary ellipsoid maintained at potential in an infinite field, is

(37)

where is the equatorial radius, and is the polar radius.

If , the figure is a circular disk of radius , and

(38)
(39)

152.] Second Case. Let , then and ,

(40)

and the equation of the hyperboloids of revolution of two sheets becomes

(41)

The quantity becomes reduced to , and each of the hyperboloids of one sheet is reduced to a pair of planes intersecting in the axis of whose equation is

(42)

This is a system of meridional planes in which is the longitude.

The quantity becomes , whence , and the equation of the family of ellipsoids is

(43)

These ellipsoids, in which the transverse axis is the axis of revolution, are called Ovary ellipsoids.

The quantity of electricity on an ovary ellipsoid maintained at a potential in an infinite field is

(44)

If the polar radius is , and the equatorial radius is ,

(45)