Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/229

Particular Cases.

151.] If $k$ is diminished till it becomes ultimately zero, the system of surfaces becomes transformed in the following manner :—

The real axis and one of the imaginary axes of each of the hyperboloids of two sheets are indefinitely diminished, and the surface ultimately coincides with two planes intersecting in the axis of $z$.

The quantity $\alpha$ becomes identical with $\theta$, and the equation of the system of meridional planes to which the first system is reduced is

${\frac {x^{2}}{(\sin \alpha )^{2}}}-{\frac {y^{2}}{(\sin \alpha )^{2}}}=0.$

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The quantity $\beta$ is reduced to

$\beta =\int {\frac {d\phi }{\sin \phi }}=\log \tan {\frac {\phi }{2}},$

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whence we find

$\sin \phi ={\frac {c}{e^{\beta }+e^{-\beta }}},\ \cos \phi ={\frac {e^{\beta }-e^{-\beta }}{e^{\beta }+e^{-\beta }}}.$

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If we call the exponential quantity ${\tfrac {1}{2}}\left(e^{\beta }+e^{-\beta }\right)$ the hyperbolic cosine of $\beta$, or more concisely the hypocosine of $\beta$, or $\cos h\beta$, and if we call ${\tfrac {1}{2}}\left(e^{\beta }-e^{-\beta }\right)$ the hyposine of $\beta$, or $\sin h\beta$, and if by the same analogy we call

	${\frac {1}{\cos h\beta }}$	the hyposecant of $\beta$, or $\sec h\beta$,
	${\frac {1}{\sin h\beta }}$	the hypocosecant of $\beta$, or $\operatorname {cosec} \ h\beta$,
	${\frac {\sin h\beta }{\cos h\beta }}$	the hypotangent of $\beta$, or $\tan h\beta$,
and	${\frac {\cos h\beta }{\sin h\beta }}$	the hypocotangent of $\beta$, or $\cot h\beta$;

then $\lambda _{2}=c\sec \ h\beta$, and the equation of the system of hyperboloids of one sheet is

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

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The quantity $\gamma$ is reduced to $\psi$, so that $\lambda _{3}=c\ \operatorname {cosec} \gamma$, and the equation of the system of ellipsoids is

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

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Ellipsoids of this kind, which are figures of revolution about their conjugate axes, are called Planetary ellipsoids.