Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/128

The reading of the torsion circle should now be adjusted, so that the coefficient of $\tau ^{\prime }$ may be as nearly as possible zero. For this purpose we must determine $\lambda$, the value of $\alpha -\beta$ when there is no torsion. This may be done by placing a non-magnetic bar of the same weight as the magnet in the stirrup, and determining $\alpha -\beta$ when there is equilibrium. Since $\tau ^{\prime }$ is small, great accuracy is required. Another method is to use a torsion bar of the same weight as the magnet, containing within it a very small magnet whose magnetic moment is ${\tfrac {1}{n}}$ of that of the principal magnet. Since $\tau$ remains the same, $\tau ^{\prime }$ will become $n\tau ^{\prime }$, and if $\zeta _{1}$ and $\zeta _{1}^{\prime }$ are the values of $\zeta$ as found by the torsion bar,

$\delta ={\tfrac {1}{2}}(\zeta _{1}+\zeta _{1}^{\prime })+{\tfrac {1}{2}}n\tau ^{\prime }(\zeta _{1}+\zeta _{1}^{\prime }-2(\beta +\lambda )).$

(12)

Subtracting this equation from (11),

$2(n-1)(\beta +\lambda )=(n+{\frac {1}{\tau ^{\prime }}})(\zeta _{1}+\zeta _{1}^{\prime })-(1+{\frac {1}{\tau ^{\prime }}})(\zeta +\zeta ^{\prime }).$

(13)

Having found the value of $\beta +\lambda$ in this way, $\beta$, the reading of

the torsion circle, should he altered till

$\zeta +\zeta ^{\prime }-2(\beta +\lambda )=0,$

(14)

as nearly as possible in the ordinary position of the apparatus.

Then, since $\tau ^{\prime }$ is a very small numerical quantity, and since its coefficient is very small, the value of the second term in the expression for $\delta$ will not vary much for small errors in the values of $\tau ^{\prime }$ and $\lambda$, which are the quantities whose values are least accurately known.

The value of $\delta$, the magnetic declination, may be found in this way with considerable accuracy, provided it remains constant during the experiments, so that we may assume $\delta ^{\prime }=\delta$.

When great accuracy is required it is necessary to take account ot the variations of $\delta$ during the experiment. For this purpose observations of another suspended magnet should be made at the some instants that the different values of $\zeta$ are observed, and if $\eta ,\eta ^{\prime }$ are the observed azimuths of the second magnet corresponding to $\zeta$and $\zeta ^{\prime }$, and if $\delta$ and $\delta ^{\prime }$ are the corresponding values of $\delta$, then

$\delta ^{\prime }-\delta =\eta ^{\prime }-\eta .$

(15)

Hence, to find the value of $\delta$ we must add to (11) a correction

${\tfrac {1}{2}}(\eta -\eta ^{\prime }).$

(15)

The declination at the time of the first observation is therefore

$\delta ={\tfrac {1}{2}}(\zeta _{1}+\zeta _{1}^{\prime }+\eta -\eta ^{\prime })+{\tfrac {1}{2}}\tau ^{\prime }(\zeta +\zeta ^{\prime }-2\beta -2\lambda ).$

(16)