Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/258

 173.] We shall apply these results to the determination of the coefficients of capacity and induction of two spheres whose radii are $$a$$ and $$b$$, and the distance of whose centres is $$c$$.

In this case

Let the sphere $$A$$ be at potential unity, and the sphere $$B$$ at potential zero.

Then the successive images of a charge $$a$$ placed at the centre of the sphere $$A$$ will be those of the actual distribution of electricity. All the images will lie on the axis between the poles and the centres of the spheres.

The values of $$u$$ and $$v$$ for the centre of the sphere $$A$$ are

$u=2\alpha,\ v=0.$

Hence we must substitute $$a$$ or $$k\tfrac{1}{\sin h\alpha}$$ for $$P$$, and 2$$a$$ for $$u$$, and $$v=0$$ in the equations, remembering that $$P$$ itself forms part of the charge of $$A$$. We thus find for the coefficient of capacity of $$A$$

$q_{aa}=k\sum_{s=0}^{s=\infty}\frac{1}{\sin h(s\varpi-\alpha)},$

for the coefficient of induction of $$A$$ on $$B$$ or of $$B$$ on $$A$$

$q_{ab}=-k\sum_{s=1}^{s=\infty}\frac{1}{\sin hs\varpi},$

and for the coefficient of capacity of $$B$$

$q_{bb}=k\sum_{s=1}^{s=\infty}\frac{1}{\sin h(\beta+s\varpi)}.$

To calculate these quantities in terms of $$a$$ and $$b$$, the radii of the spheres, and of $$c$$ the distance between their centres, we make use of the following quantities