Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/155

468.] which are found to exist where the rock near the station is magnetic, as most igneous rocks are.

Surveys of this kind can be made only in countries where magnetic instruments can be carried about and set up in a great many stations. For other parts of the world we must be content to find the distribution of the magnetic elements by interpolation between their values at a few stations at great distances from each other.

467.] Let us now suppose that by processes of this kind, or by the equivalent graphical process of constructing charts of the lines of equal values of the magnetic elements, the values of $$X$$ and $$Y$$, and thence of the potential $$V$$, are known over the whole surface of the globe. The next step is to expand $$V$$ in the form of a series of spherical surface harmonics.

If the earth were magnetized uniformly and in the same direction throughout its interior, $$V$$ would be an harmonic of the first degree, the magnetic meridians would be great circles passing through two magnetic poles diametrically opposite, the magnetic equator would be a great circle, the horizontal force would be equal at all points of the magnetic equator, and if $$H_0$$ is this constant value, the value at any other point would be $$ H = H_0 \cos l^\prime$$, where $$l^\prime$$ is the magnetic latitude. The vertical force at any point would be $$Z = 2 H_0 \sin l^\prime$$, and if $$\theta$$ is the dip, $$\tan \theta = 2 \tan l^\prime$$.

In the case of the earth, the magnetic equator is defined to be the line of no dip. It is not a great circle of the sphere.

The magnetic poles are defined to be the points where there is no horizontal force or where the dip is 90°. There are two such points, one in the northern and one in the southern regions, but they are not diametrically opposite, and the line joining them is not parallel to the magnetic axis of the earth.

468.] The magnetic poles are the points where the value of $$V$$ on the surface of the earth is a maximum or minimum, or is stationary.

At any point where the potential is a minimum the north end of the dip-needle points vertically downwards, and if a compass-needle be placed anywhere near such a point, the north end will point towards that point.

At points where the potential is a maximum the south end of the dip-needle points downwards, and the south end of the compass-needle points towards the point.

If there are $$p$$ minima of $$V$$ on the earth s surface there must be $$p - 1$$ other points, where the north end of the dip-needle points