Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/257

 $x^{2}+(y-k\cot v)^{2}=k^{2}\operatorname\{cosec\}^{2}v,$

$(x+k\cot hu)^{2}+y^{2}=k^{2}\operatorname\{cosec\}h^{2}u,$

$\cot v=\frac{x^{2}+y^{2}-k^{2}}{2ky},\ \cot hu=-\frac{x^{2}+y^{2}+k^{2}}{2kx};$|undefined

$\xi=\frac{\sqrt{2k}}{\sqrt{\cos hu-\cos v}}.$ |undefined

Since the charge of each image is proportional to its parameter, $$\xi$$, and is to be taken positively or negatively according as it is of the form $$P$$ or $$Q$$, we find

$\begin{array}{ll} P_{s}= & \frac{P\sqrt{\cos hu-\cos v}}{\sqrt{\cos h(u+2s\varpi)-\cos v}},\\ \\Q_{s}=- & \frac{P\sqrt{\cos hu-\cos v}}{\sqrt{\cos h(2\alpha-u-2s\varpi)-\cos v}},\\ \\P'_{s}= & \frac{P\sqrt{\cos hu-\cos v}}{\sqrt{\cos h(u-2s\varpi)-\cos v}},\\ \\Q_{s}=- & \frac{P\sqrt{\cos hu-\cos v}}{\sqrt{\cos h(2\beta-u+2s\varpi)-\cos v}}.\end{array}$|undefined

We have now obtained the positions and charges of the two infinite series of images. We have next to determine the total charge on the sphere $$A$$ by finding the sum of all the images within it which are of the form $$Q$$ or $$P'$$. We may write this

In the same way the total induced charge on $$B$$ is