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

$k={\frac {\sqrt {a^{4}+b^{4}+c^{4}-2b^{2}c^{2}-2c^{2}a^{2}-2a^{2}b^{2}}}{2c}};$ $\sin h\cdot \alpha ={\frac {k}{a}},\ \sin h\cdot \beta ={\frac {k}{b}}.$

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

$p=e^{\alpha }={\sqrt {{\frac {k^{2}}{a^{2}}}+1}}-{\frac {k}{a}},$ $q=e^{\beta }={\sqrt {{\frac {k^{2}}{b^{2}}}+1}}+{\frac {k}{b}},$ ${\frac {q}{p}}=r=e^{\varpi }=\left({\sqrt {{\frac {k^{2}}{a^{2}}}+1}}+{\frac {k}{a}}\right)\left({\sqrt {{\frac {k^{2}}{b^{2}}}+1}}+{\frac {k}{b}}\right).$