Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/266

The result of this investigation is that if $f$ is the diameter of the sphere, $a$ the chord of the radius of the bowl, and $r$ the chord of the distance of $P$ from the pole of the bowl, then the surface-density $\sigma$ on the inside of the bowl is

$\sigma ={\frac {V}{2\pi ^{2}f}}\left\{{\sqrt {\frac {f^{2}-a^{2}}{a^{2}-r^{2}}}}-\tan ^{-1}{\sqrt {\frac {f^{2}-a^{2}}{a^{2}-r^{2}}}}\right\},$

and the surface-density on the outside of the bowl at the same point is

$\sigma +{\frac {V}{2\pi f}}.$

In the calculation of this result no operation is employed more abstruse than ordinary integration over part of a spherical surface. To complete the theory of the electrification of a spherical bowl we only require the geometry of the inversion of spherical surfaces.

181.] Let it be required to find the surface-density induced at any point of the bowl by a quantity $q$ of electricity placed at a point $Q$, not now in the spherical surface produced.

Invert the bowl with respect to $Q$, the radius of the sphere of inversion being $R$. The bowl $S$ will be inverted into its image $S'$, and the point $P$ will have $P'$ for its image. We have now to determine the density $\sigma '$ at $P'$ when the bowl $S'$ is maintained at potential $V'$, such that $q=V'R$, and is not influenced by any external force.

The density $\sigma$ at the point $P$ of the original bowl is then

$\sigma =-{\frac {\sigma 'R^{3}}{QP^{3}}},$

this bowl being at potential zero, and influenced by a quantity $q$ of electricity placed at $Q$.

The result of this process is as follows:

Fig. 16.

Let the figure represent a section through the centre, $O$, of the sphere, the pole, $C$, of the bowl, and the influencing point $Q$. $D$ is a point which corresponds in the inverted figure to the unoccupied pole of the rim of the bowl, and may be found by the following construction.

Draw through $Q$ the chords $EQE'$ and $FQF'$, then if we suppose the radius of the sphere of inversion to be a mean proportional between the segments into which a chord is divided at $Q$, $E'F'$ will be the