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

$Q=c{\frac {V}{{\frac {\pi }{2}}-\gamma }}$

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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

$\sigma ={\frac {V}{\pi ^{2}{\sqrt {c^{2}-r^{2}}}}},$

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$Q=c{\frac {V}{\frac {\pi }{2}}}.$

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

$\alpha =\log \tan {\frac {\pi -2\theta }{4}}\ \mathrm {whence} \ \lambda _{1}=c\tan h\alpha$

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

${\frac {x^{2}}{(\tan h\alpha )^{2}}}-{\frac {y^{2}+z^{2}}{(\sec h\alpha )^{2}}}=c^{2}.$

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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

${\frac {y^{2}}{(\sin \beta )^{2}}}-{\frac {z^{2}}{(\sin \beta )^{2}}}=0.$

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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

${\frac {x^{2}}{(\cot h\gamma )^{2}}}+{\frac {y^{2}+z^{2}}{(\operatorname {cosec} \ h\gamma )^{2}}}=c^{2}.$

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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

$Q=c{\frac {V}{\gamma }}.$

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

$\gamma =\log {\frac {A+{\sqrt {A^{2}-B^{2}}}}{2B}}$

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