Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/50

We have thus found a point, fixed with respect to the magnet, such that the second term of the potential assumes the most simple form when this point is taken as origin of coordinates. This point we therefore define as the centre of the magnet, and the axis drawn through it in the direction formerly defined as the direction of the magnetic axis may be defined as the principal axis of the magnet.

We may simplify the result still more by turning the axes of y and z round that of x through half the angle whose tangent is ${\frac {P}{M-N}}$. This will cause P to become zero, and the final form of the potential may be written

$K{\frac {\xi }{r^{3}}}+{\frac {3}{2}}{\frac {(\eta ^{2}-\zeta ^{2})(M-N)}{r^{5}}}.$

(15)

This is the simplest form of the first two terms of the potential of a magnet. When the axes of y and z are thus placed they may be called the Secondary axes of the magnet.

We may also determine the centre of a magnet by finding the position of the origin of coordinates, for which the surface-integral of the square of the second term of the potential, extended over a sphere of unit radius, is a minimum.

The quantity which is to be made a minimum is, by Art. 141,

$4(l^{2}+M^{2}+N^{2}-MN-NL-LM)+3(P^{2}+Q^{2}+R^{2}).\,$

(16)

The changes in the values of this quantity due to a change of position of the origin may be deduced from equations (11) and (12). Hence the conditions of a minimum are

${\begin{matrix}2l(2L-M-N)+3nQ+3mR=0,\\2m(2M-N-L)+3lR+3nP=0,\\2n(2N-L-M)+3mP+3lQ=0.\end{matrix}}$

(17)

If we assume l = 1, m = 0, n = 0, these conditions become

$2L-M-N=0,\quad Q=0,\quad R=0,$

(18)

which are the conditions made use of in the previous investigation.

This investigation may be compared with that by which the potential of a system of gravitating matter is expanded. In the latter case, the most convenient point to assume as the origin is the centre of gravity of the system, and the most convenient axes are the principal axes of inertia through that point.

In the case of the magnet, the point corresponding to the centre of gravity is at an infinite distance in the direction of the axis,