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



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

The quantity $$\beta$$ is reduced to

whence we find

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

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

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

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