# Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/47

MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING.

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are solutions of the equation

${\displaystyle {\frac {d^{2}\psi }{dz^{2}}}+{\frac {d^{2}\psi }{d\varpi ^{2}}}-{\frac {1}{\varpi }}{\frac {d\psi }{d\varpi }}=0}$.

For putting ${\displaystyle \psi =\varpi \phi }$, this equation becomes

${\displaystyle {\frac {d^{2}\phi }{dz^{2}}}+{\frac {d^{2}\phi }{d\varpi ^{2}}}-{\frac {1}{\varpi }}{\frac {d\Phi }{d\varpi }}-{\frac {1}{\varpi ^{2}}}\Phi =0}$,

that is, ${\displaystyle \Phi \cos \phi }$ is a solution of Laplace's equation.

§ 2. Expansion of the Functions.—Let the plane through the axis of the ring and a point ${\displaystyle \mathrm {P} }$, cut the ring in the two circles whose centres are ${\displaystyle \mathrm {C} }$ and ${\displaystyle \mathrm {C} _{1}}$.

Let

${\displaystyle \mathrm {CP} =\mathrm {R} }$;${\displaystyle \quad \mathrm {C} _{1}\mathrm {P} =\mathrm {R} }$;${\displaystyle \quad \mathrm {OC} =c}$;

and the angle

${\displaystyle \mathrm {OCP} =\chi }$.

Also, for convenience, put

${\displaystyle s={\frac {\mathrm {R} }{c}}}$,

and

${\displaystyle l=\log {\frac {8c}{\mathrm {R} }}-2}$.

When ${\displaystyle \mathrm {R} }$ is less than ${\displaystyle c}$ the above integrals may be expanded in ascending powers of ${\displaystyle \mathrm {R} /c}$ or ${\displaystyle s}$.

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

${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {\left\{1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi \right\}}}}}$
${\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+{\frac {1^{2}3^{2}}{2^{2}4^{2}}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\right\}.}$

[Cayley, 'Ell. Func.,' p. 54.]