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

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

where $$c\sec\gamma$$ is the equatorial radius, and $$c\tan\gamma$$ is the polar radius.

If $$\gamma=0$$, the figure is a circular disk of radius $$c$$, and

152.] Second Case. Let $$b=c$$, then $$k=1$$ and $$k'=0$$,

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

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

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

The quantity $$\gamma$$ becomes $$\log\tan\tfrac{\pi-2\psi}{4}$$, whence $$\lambda_{3}=c\cot h\gamma,$$, and the equation of the family of ellipsoids is

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 $$V$$ in an infinite field is

If the polar radius is $$A=c\ \cot h\gamma$$, and the equatorial radius is $$B=c\ \operatorname{cosec}h\gamma$$,