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

Rh $$\mathrm{J}_4\varpi = \frac{2!}{c^3\mathrm{R}^3}\left\{\cos3\chi + \left(2\cos2\chi - \tfrac{1}{4}\cos4\chi\right)s + \left(\tfrac{61}{16}\cos\chi - \tfrac{23}{32}\cos3\chi - \tfrac{1}{32}\cos5\chi\right)s^2 +\,.\,.\,.\right\}$$

$$\mathrm{J}_5\varpi = \frac{3!}{c^4\mathrm{R}^4}\left\{\cos4\chi + \left(\tfrac{29}{12}\cos3\chi - \tfrac{1}{4}\cos5\chi\right)s +\,.\,.\,.\right\}$$

$$\mathrm{J}_6\varpi = \frac{4!}{c^5\mathrm{R}^5}\left\{\cos5\chi +\,.\,.\,.\right\}.$$

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

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

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