Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/228

Limiting Forms.

(1) When $\beta =0$ the surface is the part of the plane of $xz$ between the two branches of the hyperbola whose equation is written above, (24).

(2) When $\beta =F(k')$) the surface is the part of the plane of xy which is on the outside of the focal ellipse whose equation is

${\frac {x^{2}}{c^{2}}}+{\frac {y^{2}}{c^{2}-b^{2}}}=1.$

(25)

The Ellipsoids.

For any given ellipsoid $\gamma$ is constant. If two ellipsoids, $\gamma _{1}$ and $\gamma _{2}$, be maintained at potentials $V_{1}$ and $V_{2}$ then, for any point $\gamma$ in the space between them, we have

$V={\frac {\gamma _{1}V_{2}-\gamma _{2}V_{1}+\gamma \left(V_{1}-V_{2}\right)}{\gamma _{1}-\gamma _{2}}}.$

(26)

The surface-density at any point is

$\sigma =-{\frac {1}{4\pi }}{\frac {V_{1}-V_{2}}{\gamma _{1}-\gamma _{2}}}{\frac {cp_{3}}{P_{3}}},$

(27)

where $p_{3}$ is the perpendicular from the centre on the tangent plane, and $P_{3}$ is the product of the semi-axes.

The whole charge of electricity on either surface is

$Q_{2}=c{\frac {V_{1}-V_{2}}{\gamma _{1}-\gamma _{2}}}=-Q_{1},$

(28)

a finite quantity.

When $\gamma =F(k)$ the surface of the ellipsoid is at an infinite distance in all directions.

If we make $V_{2}=0$ and $\gamma _{2}=F(k)$, we find for the quantity of electricity on an ellipsoid maintained at potential $V$ in an infinitely extended field,

$Q=c{\frac {V}{F(k)-\gamma }}.$

(29)

The limiting form of the ellipsoids occurs when $\gamma =0$, in which case the surface is the part of the plane of $xy$ within the focal ellipse, whose equation is written above, (25).

The surface-density on the elliptic plate whose equation is (25), and whose eccentricity is $k$, is

$\sigma ={\frac {V}{2\pi {\sqrt {c^{2}-b^{2}}}}}{\frac {1}{F(k)}}{\frac {1}{\sqrt {1-{\frac {x^{2}}{c^{2}}}-{\frac {y^{2}}{c^{2}-b^{2}}}}}},$

(30)

and its charge is

$Q=c{\frac {V}{F(k)}}.$

(31)