Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/226

an elliptic integral of the first kind, which we may write according to the usual notation $F(k\theta )$.

In the same way we find

$\beta =\int _{0}^{\phi }{\frac {d\phi }{\sqrt {1-k'^{2}\cos ^{2}\phi }}}=F(k')-F(k'\phi ),$

(16)

where $Fk'$ is the complete function for modulus $k'$,

$\gamma =\int _{0}^{\psi }{\frac {d\psi }{\sqrt {1-k{}^{2}\sin ^{2}\psi }}}=F(k\psi ).$

(17)

Here $\alpha$ is represented as a function of the angle $\theta$, which is a function of the parameter $\lambda _{1}$, $\beta$ as a function of $\phi$ and thence of $\lambda _{2}$, and $\gamma$ as a function of $\psi$ and thence of $\lambda _{3}$.

But these angles and parameters may be considered as functions of $\alpha ,\ \beta ,\ \gamma$. The properties of such inverse functions, and of those connected with them, are explained in the treatise of M. Lamé on that subject.

It is easy to see that since the parameters are periodic functions of the auxiliary angles, they will be periodic functions of the quantities $\alpha ,\ \beta ,\ \gamma$: the periods of $\lambda _{1}$ and $\lambda _{3}$ are $4F(k)$ and that of $\lambda _{2}$ is $2F(k')$.

Particular Solutions.

150.] If $V$ is a linear function of $\alpha ,\ \beta$, or $\gamma$, the equation is satisfied. Hence we may deduce from the equation the distribution of electricity on any two confocal surfaces of the same family maintained at given potentials, and the potential at any point between them.

The Hyperboloids of Two Sheets.

When $\alpha$ is constant the corresponding surface is a hyperboloid of two sheets. Let us make the sign of a the same as that of $x$ in the sheet under consideration. We shall thus be able to study one of these sheets at a time.

Let $\alpha _{1}$, $\alpha _{2}$ be the values of $\alpha$ corresponding to two single sheets, whether of different hyperboloids or of the same one, and let $V_{1}$, $V_{2}$ be the potentials at which they are maintained. Then, if we make

$V={\frac {\alpha _{1}V_{2}-\alpha _{2}V_{1}+\alpha \left(V_{1}-V_{2}\right)}{\alpha _{1}-\alpha _{2}}},$

(18)

the conditions will be satisfied at the two surfaces and throughout the space between them. If we make $V$ constant and equal to $V_{1}$ in the space beyond the surface a $\alpha _{1}$ and constant and equal to $V_{2}$