Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/152

CHAPTER VIII.

ON TERRESTRIAL MAGNETISM.

465.] Our knowledge of Terrestrial Magnetism in derived from the study of the distribution of magnetic force on the earth's surface at any one time, and of the changes in that distribution at different times.

The magnetic force at any one place and time is known when its three coordinates are known. These coordinates may be given in the form of the declination or azimuth of the force, the dip or inclination to the horizon, and the total intensity.

The most convenient method, however, for investigating the general distribution of magnetic force on the earth's surface is to consider the magnitudes of the three components of the force,

$X=H\cos \delta$, directed due north,	}	(1)
$Y=H\sin \delta$, directed due west,
$Z=H\tan \theta$, directed vertically downwards,

where $H$ denotes the horizontal force, $\delta$ the declination, and $\theta$ the dip.

If $V$ is the magnetic potential of the earth's surface, and if we consider the earth a sphere of radius $a$, then

where $l$ is the latitude, and $\lambda$ the longitude, and $r$ the distance from the centre of the earth.

A knowledge of $V$ over the surface of the earth may be obtained from the observations of horizontal force alone as follows.

Let $V_{0}$ be the value of $V$ at the true north pole, then, taking the line-integral along any meridian we find,

for the value of the potential on that meridian at latitude $l$.