Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/45

Section I.—Preliminary Analysis.

§ 1. Take a fine ring in the plane of $x$, $y$, centre at the origin, and consisting of attracting matter of density $k\cos n\phi ^{\prime }$, where $\phi ^{\prime }$ is the azimuth of any point.

The potential of this ring at a point $\mathrm {P}$, whose coordinates are $r,\theta ,\phi$, is

$\int _{0}^{2\pi }{\frac {k\cos n\phi ^{\prime }\;cd\phi ^{\prime }}{\sqrt {\left\{r^{2}+c^{2}-2cr\sin \theta \cos \left(\phi -\phi ^{\prime }\right)\right\}}}}$.

Put $\phi ^{\prime }=\phi +\psi$, then the potential

$=\int _{0}^{2\pi }{\frac {ck\left(\cos n\phi \;\cos n\psi -\sin n\phi \;\sin n\psi \right)\;d\psi }{\sqrt {\left\{r^{2}+c^{2}-2cr\sin \theta \;\cos \psi \right\}}}}$

$=\cos n\phi \int _{0}^{2\pi }{\frac {ck\cos n\psi \;d\psi }{\sqrt {\left\{r^{2}+c^{2}-2cr\sin \theta \;\cos \psi \right\}}}}$

as the second integral vanishes between the limits.

Therefore

$\cos n\phi \int _{0}^{\pi }{\frac {\cos n\phi \;d\phi }{\sqrt {\left(r^{2}+c^{2}-2cr\sin \theta \;\cos \phi \right)}}}$

or

$\cos n\phi \int _{0}^{\pi }{\frac {\cos n\phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$

is a solution of Laplace's equation.