Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/361

 

In the ordinary use of the instrument the planes of the two coils are nearly at right angles to each other, so that the mutual action of the currents in the coils may be as great as possible, and the plane of the suspended coil is nearly at right angles to the magnetic meridian, so that the action of terrestrial magnetism may be as small as possible.

Let the magnetic azimuth of the plane of the fixed coil be $\alpha$, and let the angle which the axis of the suspended coil makes with the plane of the fixed coil be $\theta +\beta$, where $\beta$ is the value of this angle when the coil is in equilibrium and no current is flowing, and $\theta$ is the deflexion due to the current. The equation of equilibrium is

$Gg\gamma _{1}\gamma _{2}\cos(\theta +\beta )-Hg\gamma _{2}\sin(\theta +\beta +\alpha )-F\sin \theta =0$.

Let us suppose that the instrument is adjusted so that $\alpha$ and $\beta$ are both very small, and that $Hg\gamma _{2}$ is small compared with $F$. We have in this case, approximately,

$\tan \theta ={\frac {Gg\gamma _{1}\gamma _{2}\cos \beta }{F}}-{\frac {Hg\gamma _{2}\sin(\alpha +\beta )}{F}}-{\frac {HGg^{2}\gamma _{1}\gamma _{2}^{2}}{F^{2}}}-{\frac {G^{2}g^{2}\gamma _{1}^{2}\gamma _{2}^{2}\sin \beta }{F^{2}}}$.

If the deflexions when the signs of $\gamma _{1}$ and $\gamma _{2}$ are changed are as follows:

$\theta _{1}$	when $\gamma _{1}$ is	${}+{}$	and $\gamma _{2}$	${}+{}$,
$\theta _{2}$	{{{1}}}„	${}-{}$	{{{1}}}„	${}-{}$,
$\theta _{3}$	{{{1}}}„	${}+{}$	{{{1}}}„	${}-{}$,
$\theta _{4}$	{{{1}}}„	${}-{}$	{{{1}}}„	${}+{}$;

then we find

$\gamma _{1}\gamma _{2}={\frac {1}{4}}{\frac {F}{Gg\cos \beta }}(\tan \theta _{1}+\tan \theta _{2}-\tan \theta _{3}-\tan \theta _{4})$.

If it is the same current which flows through both coils we may put $\gamma _{1}\gamma _{2}=\gamma ^{2}$, and thus obtain the value of $\gamma$.

When the currents are not very constant it is best to adopt this method, which is called the Method of Tangents.

If the currents are so constant that we can adjust $\beta$, the angle of the torsion-head of the instrument, we may get rid of the correction for terrestrial magnetism at once by the method of sines. In this method $\beta$ is adjusted till the deflexion is zero, so that

$\theta =-\beta$.

If the signs of $\gamma _{1}$ and $\gamma _{2}$ are indicated by the suffixes of $\beta$ as before,

$F\sin \beta _{1}=-F\sin \beta _{3}=-Gg\gamma _{1}\gamma _{2}+Hg\gamma _{2}\sin \alpha$,

$F\sin \beta _{2}=-F\sin \beta _{4}=-Gg\gamma _{1}\gamma _{2}-Hg\gamma _{2}\sin \alpha$.