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



(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

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

The surface-density at any point is

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

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,

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

and its charge is