Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/55

$\mathrm {J} _{4}\varpi ={\frac {2!}{c^{3}\mathrm {R} ^{3}}}\left\{\cos 3\chi +\left(2\cos 2\chi -{\tfrac {1}{4}}\cos 4\chi \right)s+\left({\tfrac {61}{16}}\cos \chi -{\tfrac {23}{32}}\cos 3\chi -{\tfrac {1}{32}}\cos 5\chi \right)s^{2}+\,.\,.\,.\right\}$

$\mathrm {J} _{5}\varpi ={\frac {3!}{c^{4}\mathrm {R} ^{4}}}\left\{\cos 4\chi +\left({\tfrac {29}{12}}\cos 3\chi -{\tfrac {1}{4}}\cos 5\chi \right)s+\,.\,.\,.\right\}$

$\mathrm {J} _{6}\varpi ={\frac {4!}{c^{5}\mathrm {R} ^{5}}}\left\{\cos 5\chi +\,.\,.\,.\right\}.$

These five sets of formulas will be referred to as (A), (B), (C), (D), (E).

Section II.—The Potential of an Anchor Ring at an External Point.

§ 4. The potential of an anchor ring at a point on its axis may be easily found in several ways. One simple method is to divide the ring into elements by spheres, having the given point as centre.

Let $\mathrm {QAR}$ be a circle which by revolution round $\mathrm {OP}$ generates an anchor ring. Let $\mathrm {C}$ be its centre, and let $\mathrm {OC}$ be perpendicular to $\mathrm {OP}$.

With centre $\mathrm {P}$ describe circles, dividing the circle $\mathrm {QAR}$ into elements; let $\mathrm {QR}$ and $\mathrm {Q} ^{\prime }\mathrm {R} ^{\prime }$ be two of these circles.

By revolution of the figure round $\mathrm {OP}$ we obtain an anchor ring divided into elements.

Let

$\mathrm {AC} =a.\qquad \mathrm {OC} =c.\qquad \mathrm {PC} =\mathrm {R} .\quad \mathrm {PQ} =\rho .$

$\angle \,\,\mathrm {PCQ} =\psi \qquad \angle \,\,\mathrm {OCP} =\alpha .$