Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/48

 Rh In this expression l, m, n are the direction-cosines of the axis of the magnet, and K is the magnetic moment of the magnet. If ε is the angle which the axis of the magnet makes with the direction of the magnetic force $$\mathfrak{H}$$, the value of W may be written

If the magnet is suspended so as to be free to turn about a vertical axis, as in the case of an ordinary compass needle, let the azimuth of the axis of the magnet be $$\phi$$, and let it be inclined $$\theta$$ to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is $$\delta$$ and dip $$\zeta$$, then

whence W K$ (cos cos0 cos(&amp;lt; 8) + sin &amp;lt;&amp;gt;in 0). (12)

The moment of the force tending to increase φ by turning the magnet round a vertical axis is

On the Expansion of the Potential of a Magnet in Solid Harmonics.

391.] Let V be the potential due to a unit pole placed at the point (ξ, η, ζ). The value of V at the point x, y, z is

This expression may be expanded in terms of spherical harmonics, with their centre at the origin. We have then

To determine the value of the potential energy when the magnet is placed in the field of force expressed by this potential, we have to integrate the expression for W in equation (3) with respect to x, y and z considering ξ, η, ζ as constants.

If we consider only the terms introduced by V0, V1 and V2 the result will depend on the following volume-integrals,