Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/337

 

${\overline {P}}={}$	$P_{0}+{\frac {1}{24}}\left(x^{2}{\frac {d^{2}P_{0}}{dx^{2}}}+y^{2}{\frac {d^{2}P_{0}}{dy^{2}}}\right)$
	${}+{\frac {1}{960}}\left(x^{4}{\frac {d^{4}P_{0}}{dx^{4}}}+y^{4}{\frac {d^{4}P_{0}}{dy^{4}}}\right)+{\frac {1}{576}}x^{2}y^{2}{\frac {d^{4}P_{0}}{dx^{2}\,dy^{2}}}+\mathrm {\&c.}$

In the case of the coil, let the outer and inner radii be $A+{\frac {1}{2}}\xi$, and $A-{\frac {1}{2}}\xi$ respectively, and let the distance of the planes of the windings from the origin lie between $B+{\frac {1}{2}}\eta$ and $B-{\frac {1}{2}}\eta$, then the breadth of the coil is $\eta$, and its depth $\xi$, these quantities being small compared with $A$ or $C$.

In order to calculate the magnetic effect of such a coil we may write the successive terms of the series as follows:—

$G_{0}=\pi {\frac {B}{C}}\left(1+{\frac {1}{24}}{\frac {2A^{2}-B^{2}}{C^{4}}}\xi ^{2}-{\frac {1}{8}}{\frac {A^{2}}{C^{4}}}\eta ^{2}\right)$,

$G_{1}=2\pi {\frac {A^{2}}{C^{3}}}\left(1+{\frac {1}{24}}\left({\frac {2}{A^{2}}}-15{\frac {B^{2}}{C^{4}}}\right)\xi ^{2}+{\frac {1}{8}}{\frac {4B^{2}-A^{2}}{C^{4}}}\eta ^{2}\right)$,

$G_{2}=3\pi {\frac {A^{2}B}{C^{5}}}\left(1+{\frac {1}{21}}\left({\frac {2}{A^{2}}}-{\frac {25}{C^{2}}}+{\frac {35A^{2}}{C^{4}}}\right)\xi ^{2}+{\frac {5}{24}}{\frac {4B^{2}-3A^{2}}{C^{4}}}\eta ^{2}\right)$,

$G_{3}=4\pi {\frac {A^{2}(B^{2}-{\frac {1}{4}}A^{2})}{C^{7}}}+{\frac {\pi }{24}}{\frac {\xi ^{2}}{C^{11}}}\left\{C^{4}(8B^{2}-12A^{2})+35A^{2}B^{2}(5A^{2}-4B^{2})\right\}$

${}+{\frac {\pi }{24}}{\frac {\eta ^{2}}{C^{11}}}\left\{3A^{2}C^{2}(5A^{2}-44B^{2})+63A^{2}B^{2}(4B^{2}-A^{2})\right\}$,

&c., &c.;

$g_{1}=\pi a^{2}$	${}+{\frac {1}{12}}\pi \xi ^{2}$,
$g_{2}=2\pi a^{2}b$	${}+{\frac {1}{6}}\pi b\xi ^{2}$,
$g_{2}=3\pi a^{2}(b^{2}-{\frac {1}{4}}a^{2})$	${}+{\frac {\pi }{8}}\xi ^{2}(2b^{2}-3a^{2})+{\frac {\pi }{4}}\eta ^{2}a^{2}$,
&c., &c.

The quantities $G_{0}$, $G_{1}$, $G_{2}$, &c. belong to the large coil. The value of $\omega$ at points for which $r$ is less than $C$ is

$\omega =-2\pi +2G_{0}-G_{1}rQ_{1}(\theta )-G_{2}r^{2}Q_{2}(\theta )-\mathrm {\&c.}$

The quantities $g_{1}$, $g_{2}$, &c. belong to the small coil. The value of $\omega ^{\prime }$ at points for which $r$ is greater than $c$ is

$\omega ^{\prime }=g_{1}{\frac {1}{r^{2}}}Q_{1}(\theta )+g_{2}{\frac {1}{r^{3}}}Q_{2}(\theta )+\mathrm {\&c.}$

The potential of the one coil with respect to the other when the total current through the section of each coil is unity is

$M=G_{1}g_{1}Q_{1}(\theta )+G_{2}g_{2}Q_{2}(\theta )+\mathrm {\&c.}$

To find $M$ by Elliptic Integrals.

701.] When the distance of the circumferences of the two circles