Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/398

The equation of motion of the suspended magnet is

${\frac {d^{2}T}{d{\dot {\phi }}\,dt}}-{\frac {dT}{d\phi }}+{\frac {dV}{d\phi }}=0$,

(10)

whence

$A{\ddot {\phi }}-MG\gamma \cos(\theta -\phi )+MH(\sin \phi +\tau (\phi -a))=0$.

(11)

Substituting the value of $\gamma$, and arranging the terms according to the functions of multiples of $\theta$, then we know from observation that

$\phi =\phi _{0}+be^{-lt}\cos nt+c\cos 2(\theta -\beta )$,

(12)

where $\phi _{0}$ is the mean value of $\phi$, and the second term expresses the free vibrations gradually decaying, and the third the forced vibrations arising from the variation of the deflecting current.

The value of $n$ in equation (12) is ${\frac {HM}{A}}\sec \phi$. That of $c$, the amplitude of the forced vibrations, is ${\frac {1}{4}}{\frac {n^{2}}{\omega ^{2}}}\sin \phi$. Hence, when the coil makes many revolutions during one free vibration of the magnet, the amplitude of the forced vibrations of the magnet is very small, and we may neglect the terms in (11) which involve $c$.

Beginning with the terms in (11) which do not involve $\theta$, we find ��

cos $o + I* W Sin (f> n ) H -- ^r- R

��(sin + T(^ -a)). (13)

Remembering that ${\dot {\phi }}$ is small, and that $L$ is generally small compared with $Gg$, we find as a sufficiently approximate value of $R$,

$R={\frac {Gg\omega }{2\tan \phi _{0}\left(1+\tau {\frac {\phi -a}{\sin \phi }}\right)}}\left\{1+{\frac {GM}{gH}}\sec \phi -{\frac {2L}{gG}}\left({\frac {2L}{Gg}}-1\right)\tan ^{2}\phi \right\}$.

(14)

766.] The resistance is thus determined in electromagnetic measure in terms of the velocity $\omega$ and the deviation $\phi$. It is not necessary to determine $H$, the horizontal terrestrial magnetic force, provided it remains constant during the experiment.

To determine ${\frac {M}{H}}$ we must make use of the suspended magnet to deflect the magnet of the magnetometer, as described in Art. 454. In this experiment $M$ should be small, so that this correction be comes of secondary importance.

For the other corrections required in this experiment see the Report of the British Association for 1863, p. 168.