Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/332

Let

$CH=c$,

$OC=b=c\cos a$,

$OH=a=c\sin \alpha$.

Let $A$ be the pole of the sphere, and $Z$ any point on the axis, and let $CZ=z$.

Let $R$ be any point in space, and let $CR=r$, and $ACR=\theta$.

Let $P$ be the point when $CR$ cuts the sphere.

The magnetic potential due to the circular current is equal to that due to a magnetic shell of strength unity bounded by the current. As the form of the surface of the shell is indifferent, provided it is bounded by the circle, we may suppose it to coincide with the surface of the sphere.

We have shewn in Art. 670 that if $P$ is the potential due to a stratum of matter of surface-density unity, spread over the surface of the sphere within the small circle, the potential due to a magnetic shell of strength unity and bounded by the same circle is

We have in the first place, therefore, to find $P$.

Let the given point be on the axis of the circle at $Z$, then the part of the potential at $Z$ due to an element $dS$ of the spherical surface at $P$ is

This may be expanded in one of the two series of spherical harmonics,

		⁠	${\frac {dS}{c}}\left\{Q_{0}+Q_{1}{\frac {z}{c}}+\mathrm {\&c.} +Q_{i}{\frac {z^{i}}{c^{i}}}+\mathrm {\&c.} \right\}$,
	or		${\frac {dS}{z}}\left\{Q_{0}+Q_{1}{\frac {c}{z}}+\mathrm {\&c.} +Q^{i}{\frac {z^{i}}{z^{i}}}+\mathrm {\&c.} \right\}$,

the first series being convergent when $z$ is less than $c$, and the second when $z$ is greater than $c$.

and integrating with respect to $\phi$ between the limits $0$ and $2\pi$, and with respect to $\mu$ between the limits $\cos \alpha$ and $1$, we find

			$P=2\pi c\left\{\int _{\mu }^{1}Q_{0}d\mu +\mathrm {\&c.} +{\frac {z^{i}}{c^{i}}}\int _{\mu }^{1}Q_{i}d\mu \right\}$,		(1)
	or	⁠	$P^{\prime }=2\pi {\frac {c^{2}}{z}}\left\{\int _{\mu }^{1}Q_{0}d\mu +\mathrm {\&c.} +{\frac {c^{i}}{z^{i}}}\int _{\mu }^{1}Q_{i}d\mu \right\}$.		(1′)

By the characteristic equation of $Q_{i}$,

$i(i+1)Q_{i}+{\frac {d}{d\mu }}\left[(1-\mu ^{2}){\frac {dQ_{i}}{d\mu }}\right]=0$.