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

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

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

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

$\xi ={\frac {\sqrt {2k}}{\sqrt {\cos hu-\cos v}}}.$^[1]

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

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

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

$P{\sqrt {\cos hu-\cos v}}\sum _{s=1}^{s=\infty }{\frac {1}{\sqrt {\cos h(u-2s\varpi )-\cos v}}},$ $-P{\sqrt {\cos hu-\cos v}}\sum _{s=0}^{s=\infty }{\frac {1}{\sqrt {\cos h(2\alpha -u-2s\varpi )-\cos v}}},$

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

$P{\sqrt {\cos hu-\cos v}}\sum _{s=1}^{s=\infty }{\frac {1}{\sqrt {\cos h(u+2s\varpi )-\cos v}}},$ $-P{\sqrt {\cos hu-\cos v}}\sum _{s=1}^{s=\infty }{\frac {1}{\sqrt {\cos h(2\beta -u+2s\varpi )-\cos v}}},$

1. In these expressions we must remember that

   $2\cos hu=e^{u}+e^{-u},\ 2\sin hu=e^{u}-e^{-u},$

   and the other functions of $u$ are derived from these by the same definitions as the corresponding trigonometrical functions.
   The method of applying dipolar coordinates to this case was given by Thomson in Liouville's Journal for 1847. See Thomson's reprint of Electrical Papers, 211, 212. In the text I have made use of the investigation of Prof. Betti, Nuovo Cimento, vol. xx, for the analytical method, but I have retained the idea of electrical images as used by Thomson in his original investigation, Phil. Mag., 1853.