Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/135

$n$ negative passages have been observed. If the vibrations are too rapid to allow of every consecutive passage being observed, every third or every fifth passage is observed, care being taken that the observed passages are alternately positive and negative.

Let the observed times of passage be $T_{1},T_{2},T_{2n+1}$, then if we put

${\frac {1}{n}}\left({\frac {1}{2}}T_{1}+T_{3}+T_{5}+\&{\text{c.}}+T_{2n-1}+{\frac {1}{2}}T_{2n+1}\right)=T_{n+1},\quad {\frac {1}{n}}(T_{2}+T_{4}\&{\text{c.}}\qquad +T_{2n})=T'_{n+1}$;

then $T_{n+1}$ is the mean time of the positive passages, and ought to agree with $T'{n+l}$, the mean time of the negative passages, if the point has been properly chosen. The mean of these results is to be taken as the mean time of the middle passage.

After a large number of vibrations have taken place, but before the vibrations have ceased to be distinct and regular, the observer makes another series of observations, from which he deduces the mean time of the middle passage of the second series.

By calculating the period of vibration either from the first series of observations or from the second, he ought to be able to be certain of the number of whole vibrations which have taken place in the interval between the time of middle passage in the two series. Dividing the interval between the mean times of middle passage in the two series by this number of vibrations, the mean time of vibration is obtained.

The observed time of vibration is then to be reduced to the time of vibration in infinitely small arcs by a formula of the same kind as that used in pendulum observations, and if the vibrations are found to diminish rapidly in amplitude, there is another correction for resistance, see Art. 740. These corrections, however, are very small when the magnet hangs by a fibre, and when the arc of vibration is only a few degrees.

The equation of motion of the magnet is

$A{\frac {d^{2}\theta }{dt^{2}}}+MH\sin \theta +MH\tau '(\theta -\gamma )=0$

where $\theta$ is the angle between the magnetic axis and the direction of the force $H$, $A$ is the moment of inertia of the magnet and suspended apparatus, $M$ is the magnetic moment of the magnet, H the intensity of the horizontal magnetic force, and $MH\tau '$ the coefficient of torsion: $\tau '$ is determined as in Art. 452, and is a very small quantity. The value of for equilibrium is

$\theta _{0}={\frac {\tau '\gamma }{1+\tau '}}$,a very small angle,