Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/158

If we take the case in which one of the surfaces, say $S_{2}$, surrounds the rest at an infinite distance, we have the ordinary case of conductors in an infinite region; and if we make $E_{2}=-E_{1}$, and $E=0$ for all the other surfaces, we have $V_{2}=0$ at infinity, and $p_{11}$ is not greater than ${\tfrac {2Q''}{E_{1}}}$.

In the very important case in which the electrical action is entirely between two conducting surfaces $S_{1}$ and $S_{2}$, of which $S_{2}$ completely surrounds $S_{1}$ and is kept at potential zero, we have $E_{1}=-E_{2}$ and $q_{11}p_{11}=1$.

Hence in this case we have

$q_{11}\ \mathrm {not\ less\ than} \ {\frac {E_{1}}{2Q''}};$

(18)

and we had before

$q_{11}\ \mathrm {not\ greater\ than} \ 2Q';$

(19)

so that we conclude that the true value of $q_{11}$, the capacity of the internal conductor, lies between these values.

This method of finding superior and inferior limits to the values of these coefficients was suggested by a memoir 'On the Theory of Resonance,' by the Hon. J. W. Strutt, Phil. Trans., 1871. See Art. 308.