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

and we had before

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.