Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/43

II. The Potential of an Anchor Ring.

By F. W. Dyson, B.A., Fellow of Trinity College, Cambridge.

Communicated by Professor J. J. Thomson, F.R.S.

Received March, 19—Read May 5, 1892.

Introduction.

In this Paper I have developed a method of dealing with questions connected with Anchor Rings.

If $r,\theta ,\phi$ be the coordinates of any point outside an anchor ring, whose central circle of radius $c$, then

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

is a solution of Laplace's equation, finite at all external points and vanishing at infinity. Let this be called $\mathrm {I}$. Then $d\mathrm {I} /dz$ is another solution; and two sets of solutions may be found by differentiating $\mathrm {I}$ and $d\mathrm {I} /dz$ any number of times with respect to $c$. These solutions are symmetrical with respect to the axis of the ring. In the first set $z$ is involved only in even powers; in the second set in odd powers.

Take a section through the axis $\mathrm {O} z$ of the ring and the point $\mathrm {P} \left(r,\theta \right)$ cutting the central circle of the ring in $\mathrm {C}$.

Let $\mathrm {CP} =\mathrm {R}$ and $\angle \,\,\mathrm {OCP} =\chi$.

When $\mathrm {R}$ is less than $c$, the integral

$\int _{0}^{\pi }{\frac {d\phi }{\sqrt {\left(r^{3}+c^{2}-2cr\sin \theta \,\cos \phi \right)}}}$