Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/141

is zero, that is, when the weight of the suspended mass is equally borne by the two wires.

We may adjust the tensions of the wires to equality by observing the time of vibration, and making it a minimum, or we may obtain a self-acting adjustment by attaching the ends of the wires, as in Fig. 16, to a pulley, which turns on its axis till the tensions are equal.

The distance of the upper ends of the suspension wires is regulated by means of two other pullies. The distance between the lower ends of the wires is also capable of adjustment.

By this adjustment of the tension, the couple arising from the tensions of the wires becomes

$L={\frac {1}{4}}{\frac {ab}{h}}mg\sin(\alpha -\beta )$.

The moment of the couple arising from the torsion of the wires is of the form

$\tau (\gamma -\beta )$,

where $\tau$ is the sum of the coefficients of torsion of the wires.

The wires ought to be without torsion when $\alpha =\beta$, we may then make $\gamma =\alpha$.

The moment of the couple arising from the horizontal magnetic force is of the form

$MH\sin(\delta -\theta )$,

where $\delta$ is the magnetic declination, and is the azimuth of the axis of the magnet. We shall avoid the introduction of unnecessary symbols without sacrificing generality if we assume that the axis of the magnet is parallel to BB', or that $\beta =\theta$.

The equation of motion then becomes

There are three principal positions of this apparatus.

(1) When $\alpha$ is nearly equal to $\delta$. If $T_{1}$ is the time of a complete oscillation in this position, then

(2) When $\alpha$ is nearly equal to $\delta +\pi$. If $T_{2}$ is the time of a complete oscillation in this position, the north end of the magnet being now turned towards the south,

The quantity on the right-hand of this equation may be made