Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/242



If $$A$$ and $$B$$ are two points, $$A'$$ and $$B'$$ their images, $$O$$ being the centre of inversion, and $$R$$ the radius of the sphere of inversion,

$OA\cdot OA'=R^{2}=OB\cdot OB'$

Hence the triangles $$OAB$$, $$OB'A'$$ are similar, and

$AB:A'B'::OA:OB'::OA\cdot OB:R^{2}.$

If a quantity of electricity $$e$$ be placed at $$A$$, its potential at $$B$$ will be

$V=\frac{e}{AB}$

If $$e'$$ be placed at $$A'$$ its potential at $$B'$$ will be

$V'=\frac{e'}{A'B'}.$

In the theory of electrical images

$e:e'::OA:R::R:OA'.$

Hence

or the potential at $$B$$ due to the electricity at $$A$$ is to the potential at the image of $$B$$ due to the electrical image of $$A$$ as $$R$$ is to $$OB$$.

Since this ratio depends only on $$OB$$ and not on $$OA$$, the potential at $$B$$ due to any system of electrified bodies is to that at $$B'$$ due to the image of the system as $$R$$ is to $$OB$$.

If $$r$$ be the distance of any point $$A$$ from the centre, and $$r'$$ that of its image $$A'$$, and if $$e$$ be the electrification of $$A$$, and $$e'$$ that of $$A'$$, also if $$L, S, K$$ be linear, superficial, and solid elements at $$A$$, and $$L', S', K'$$ their images at $$A'$$, and $$\lambda,\sigma,\rho,\lambda',\sigma',\rho',$$ the corresponding line-surface and volume-densities of electricity at the two points, $$V$$ the potential at $$A$$ due to the original system, and $$V'$$ the potential at $$A'$$ due to the inverse system, then

{{MathForm2|(18) |$$\left.\begin{array}{c} \frac{r'}{r}=\frac{L'}{L}=\frac{R^{2}}{r^{2}}=\frac{r'^{2}}{r^{2}},\ \frac{S'}{S}=\frac{R^{4}}{r^{4}}=\frac{r'^{4}}{R^{4}},\ \frac{K'}{K}=\frac{R^{6}}{r^{6}}=\frac{r'^{6}}{R^{6}};\\ \\\frac{e'}{e}=\frac{R}{r}=\frac{r'}{R},\ \frac{\lambda'}{\lambda}=\frac{r}{R}=\frac{R}{r'},\\ \\\frac{\sigma'}{\sigma}=\frac{r^{3}}{R^{3}}=\frac{R^{3}}{r'^{3}},\ \frac{\rho'}{\rho}=\frac{r^{5}}{R^{5}}=\frac{R^{5}}{r'^{5}}\\ \\\frac{V'}{V}=\frac{r}{R}=\frac{R}{r'}.\end{array}\right\} $$}}

If in the original system a certain surface is that of a conductor,