Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/142

as small as we please by diminishing $a$ or $b$, but it must not be made negative, or the equilibrium of the magnet will become unstable. The magnet in this position forms an instrument by which small variations in the direction of the magnetic force may be rendered sensible.

For when $\delta -\theta$ is nearly equal to $\pi$, $\sin(\delta -\theta )$ is nearly equal to $\theta -\delta$, and we find

By diminishing the denominator of the fraction in the last term we may make the variation of $\theta$ very large compared with that of $\delta$. We should notice that the coefficient of $\delta$ in this expression is negative, so that when the direction of the magnetic force turns in one direction the magnet turns in the opposite direction.

(3) In the third position the upper part of the suspension-apparatus is turned round till the axis of the magnet is nearly perpendicular to the magnetic meridian.

If we make

the equation of motion may be written

If there is equilibrium when $H=H_{0}$ and $\theta ^{\prime }=0$,

and if $H$ is the value of the horizontal force corresponding to a small angle $\theta ^{\prime }$,

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In order that the magnet may be in stable equilibrium it is necessary that the numerator of the fraction in the second member should be positive, but the more nearly it approaches zero, the more sensitive will be the instrument in indicating changes in the value of the intensity of the horizontal component of terrestrial magnetism.

The statical method of estimating the intensity of the force depends upon the action of an instrument which of itself assumes