Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/232

$\left.{\begin{array}{rl}x=&\lambda +\mu +\nu -c-b,\\\\y^{2}=&4{\frac {(b-\lambda )(\mu -b)(\nu -b)}{c-b}},\\\\z^{2}=&4{\frac {(c-\lambda )(c-\mu )(\nu -c)}{c-b}};\end{array}}\right\}$

(50)

$\left.{\begin{array}{l}\lambda ={\frac {1}{2}}(b+c)-{\frac {1}{2}}(c-v)\cos h\alpha ,\\\mu ={\frac {1}{2}}(b+c)-{\frac {1}{2}}(c-b)\cos \beta ,\\\nu ={\frac {1}{2}}(b+c)+{\frac {1}{2}}(c-b)\cos h\gamma ;\end{array}}\right\}$

(51)

$\left.{\begin{array}{l}x={\frac {1}{2}}(b+c)+{\frac {1}{2}}(c-b)(\cos h\gamma -\cos \beta -\cos h\alpha ).\\\\y=2(c-b)\sin h{\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\cos h{\frac {\gamma }{2}},\\\\z=2(c-b)\cos h{\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\sin h{\frac {\gamma }{2}}.\end{array}}\right\}$

(52

When $b=c$ we have the case of paraboloids of revolution about the axis of $x$, and

${\begin{array}{l}x=a\left(e^{2\alpha }-e^{2\gamma }\right),\\y=2ae^{\alpha +\gamma }\cos \beta ,\\z=2ae^{\alpha +\gamma }\sin \beta .\end{array}}$

(53)

The surfaces for which $\beta$ is constant are planes through the axis, $\beta$ being the angle which such a plane makes with a fixed plane through the axis.

The surfaces for which $\alpha$ is constant are confocal paraboloids. When $\alpha =0$ the paraboloid is reduced to a straight line terminating at the origin.

We may also find the values of $\alpha ,\ \beta ,\ \gamma$ in terms of $r,\theta$ and $\phi$, the spherical polar coordinates referred to the focus as origin, and the axis of the parabolas as axis of the sphere,

${\begin{array}{l}\alpha =\log \left(r^{\frac {1}{2}}\cos {\frac {1}{2}}\theta \right),\\\beta =\phi ,\\\gamma =\log \left(r^{\frac {1}{2}}\sin {\frac {1}{2}}\theta \right).\end{array}}$

(54)

We may compare the case in which the potential is equal to $\alpha$, with the zonal solid harmonic $r_{i}Q_{i}$. Both satisfy Laplace’s equation, and are homogeneous functions of x, y, z, but in the case derived from the paraboloid there is a discontinuity at the axis, and $i$ has a value not differing by any finite quantity from zero.

The surface-density on an electrified paraboloid in an infinite field (including the case of a straight line infinite in one direction) is inversely as the distance from the focus, or, in the case of the line, from the extremity of the line.