16
ELEMENTARY THEORY OF MAGNETISM.
[391.
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
, the value of W may be written
|
 | (9) |
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
, and let it be inclined
to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is
and dip
, then
|
 | (10) |
|
 | (11) |
|
whence  | (12) |
The moment of the force tending to increase
by turning the
magnet round a vertical axis is
|
 | (13) |
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
|
 | (1) |
This expression may be expanded in terms of spherical harmonics,
with their centre at the origin. We have then
|
&c., | (2) |
when |
r being the distance of (ξ, η, ζ) from the origin,
| (3) |
|
| (4) |
|
| (5) |
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,