Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/259

 We may now write the hyperbolic sines in terms of $$p, q, r$$; thus

$\begin{array}{ll} q_{aa}= & \sum_{s=0}^{s=\infty}\frac{2k}{\frac{r^{s}}{p}-\frac{p}{r^{s}}},\\ q_{ab}=- & \sum_{s=1}^{s=\infty}\frac{2k}{r^{s}-\frac{1}{r^{s}}},\\ \\q_{bb}= & \sum_{s=0}^{s=\infty}\frac{2k}{qr^{s}-\frac{1}{qr^{s}}}.\end{array}$|undefined

Proceeding to the actual calculation we find, either by this process or by the direct calculation of the successive images as shewn in Sir W. Thomson's paper, which is more convenient for the earlier part of the series,

$\begin{array}{l} q_{aa}=a+\frac{a^{2}b}{c^{2}-b^{2}}+\frac{a^{3}b^{2}}{\left(c^{2}-b^{2}+ac\right)\left(c^{2}-b^{2}-ac\right)}+etc.,\\ \\q_{ab}=-\frac{ab}{c}-\frac{a^{2}b^{2}}{c\left(c^{2}-a^{2}-b^{2}\right)}-\frac{a^{3}b^{3}}{c\left(c^{2}-a^{2}-b^{2}+ab\right)\left(c^{2}-a^{2}-b^{2}-ab\right)}-etc.\\ \\q_{bb}=b+\frac{ab^{2}}{c^{2}-a^{2}}+\frac{a^{2}b^{3}}{\left(c^{2}-a^{2}+bc\right)\left(c^{2}-a^{2}-bc\right)}+etc.\end{array}$|undefined

174.] We have then the following equations to determine the charges $$E_a$$ and $$E_b$$ of the two spheres when electrified to potentials $$V_a$$ and $$V_b$$ respectively,

If we put

$q_{aa}q_{bb}-q_{ab}^{2}=D=\frac{1}{D'},$

and

$p_{aa}=q_{bb}D',\ p_{ab}=-q_{ab}D',\ p_{bb}=q_{aa}D',$

whence

$p_{aa}p_{bb}-p_{ab}^{2}=D';$

then the equations to determine the potentials in terms of the charges are

and $$p_{aa}, p_{ab}$$, and $$p_{bb}$$ are the coefficients of potential.

The total energy of the system is, by Art. 85,

$\begin{array}{ll} Q & =\frac{1}{2}\left(E_{a}V_{a}+E_{b}V_{b}\right),\\ \\ & =\frac{1}{2}\left(V_{a}^{2}q_{aa}+2V_{a}V_{b}q_{ab}+V_{b}^{2}q_{bb}\right),\\ \\ & =\frac{1}{2}\left(E_{a}^{2}p_{aa}+2E_{a}E_{b}p_{ab}+E_{b}^{2}p_{bb}\right).\end{array}$