Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/145

461.] Let $$p$$ be the perpendicular from the centre of inertia on the plane on which the axis rolls, then $$p$$ will be a function of $$\theta$$, whatever be the shape of the rolling surfaces. If both the rolling sections of the ends of the axis are circular, Rh where $$a$$ is the distance of the centre of inertia from the line joining the centres of the rolling sections, and $$\alpha$$ is the angle which this line makes with the line of collimation.

If $$M$$ is the magnetic moment, $$m$$ the mass of the magnet, and $$g$$ the force of gravity, $$I$$ the total magnetic force, and $$i$$ the dip, then, by the conservation of energy, when there is stable equilibrium, Rh must be a maximum with respect to $$\theta$$, or if the ends of the axis are cylindrical.

Also, if $$T$$ be the time of vibration about the position of equilibrium, Rh where $$A$$ is the moment of inertia of the needle about its axis of rotation.

In determining the dip a reading is taken with the dip circle in the magnetic meridian and with the graduation towards the west.

Let $$\theta_1$$ be this reading, then we have Rh

The instrument is now turned about a vertical axis through 180°, so that the graduation is to the east, and if $$\theta_2$$ is the new reading,

Rh

Taking (6) from (5), and remembering that $$\theta_1$$ is nearly equal to $$i$$ and $$\theta_2$$ nearly equal to $$\pi - i$$, and that $$\lambda$$ is a small angle, such that $$mga\lambda$$ may be neglected in comparison with $$MI$$,

Rh

Now take the magnet from its bearings and place it in the deflexion apparatus, Art. 453, so as to indicate its own magnetic moment by the deflexion of a suspended magnet, then Rh where $$D$$ is the tangent of the deflexion. VOL. II.