Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/333

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695.]

SOLID ANGLE SUBTENDED BY A CIRCLE.

301

Hence

\int _{\mu }^{i}Q_{i}d\mu ={\frac {1-\mu ^{2}}{i(i+1)}}{\frac {dQ_{i}}{d\mu }}

.

(2)

This expression fails when $i=0$ , but since $Q_{0}=1$ ,

\int _{\mu }^{1}Q_{0}d\mu =1-\mu

.

(3)

As the function ${\frac {dQ_{i}}{d\mu }}$ occurs in every part of this investigation we shall denote it by the abbreviated symbol $Q_{i}^{\prime }$ . The values of $Q_{i}^{\prime }$ corresponding to several values of $i$ are given in Art. 698.

We are now able to write down the value of $P$ for any point $R$ , whether on the axis or not, by substituting $r$ for $z$ , and multiplying each term by the zonal harmonic of $\theta$ of the same order. For $P$ must be capable of expansion in a series of zonal harmonics of $\theta$ with proper coefficients. When $\theta =0$ each of the zonal harmonics becomes equal to unity, and the point $R$ lies on the axis. Hence the coefficients are the terms of the expansion of $P$ for a point on the axis. We thus obtain the two series

			$P$	${}=2\pi c\left\{1-\mu +\mathrm {\&c.} +{\frac {1-\mu ^{2}}{i(i+1)}}{\frac {r^{i}}{c^{i}}}Q_{i}^{\prime }(\alpha )Q_{i}(\theta )\right\}$ ,		(4)
			or⁠ $P^{\prime }$	${}=2\pi {\frac {c^{2}}{r}}\left\{1-\mu +\mathrm {\&c.} +{\frac {1-\mu ^{2}}{i(i+1)}}{\frac {c^{i}}{r^{\prime }}}Q_{i}^{\prime }(\alpha )Q_{i}(\theta )\right\}$ .		(4′)

695.] We may now find $\omega$ , the magnetic potential of the circuit, by the method of Art. 670, from the equation

\omega ={\frac {1}{c}}{\frac {d}{dr}}(Pr)

.

(5)

We thus obtain the two series

		$\omega$	${}=-2\pi \left\{1-\cos \alpha +\mathrm {\&c.} +{\frac {\sin ^{2}\alpha }{i}}{\frac {r^{i}}{c^{i}}}Q_{i}^{\prime }(\alpha )Q_{i}(\theta )+\mathrm {\&c.} \right\}$ ,		(6)
		or⁠ $\omega ^{\prime }$	${}=2\pi \sin ^{2}\alpha \left\{{\frac {1}{2}}{\frac {c^{2}}{r^{2}}}Q_{1}^{\prime }(\alpha )Q_{1}(\theta )+\mathrm {\&c.} +{\frac {1}{i+1}}{\frac {c^{i+1}}{r^{i+1}}}Q_{i}^{\prime }(\alpha )Q_{i}(\theta )\right\}$ .		(6′)

The series (6) is convergent for all values of $r$ less than $c$ , and the series (6′) is convergent for all values of $r$ greater than $c$ . At the surface of the sphere, where $r=c$ , the two series give the same value for $\omega$ when $\theta$ is greater than $\alpha$ , that is, for points not occupied by the magnetic shell, but when $\theta$ is less than $\alpha$ , that is, at points on the magnetic shell,