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

Rh II. The Potential of an Anchor Ring.

By, B.A., Fellow of Trinity College, Cambridge.

Communicated by Professor, F.R.S.

Received March, 19—Read May 5, 1892.

Introduction.

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

$\int_0^\pi\frac{d\phi}{\sqrt{\left(r^2 + c^2 - 2cr\sin\theta\,\cos\phi\right)}}$|undefined

is a solution of '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

$\int_0^{\pi}\frac{d\phi}{\sqrt{\left(r^3 + c^2 - 2cr\sin\theta\,\cos\phi\right)}}$|undefined Rh