Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/153

466.] Thus the potential may be found for any point on the earth's surface provided we know the value of $$X$$, the northerly component at every point, and $$V_0$$, the value of $$V$$ at the pole.

Since the forces depend not on the absolute value of $$V$$ but on its derivatives, it is not necessary to fix any particular value for $$V_0$$.

The value of $$V$$ at any point may be ascertained if we know the value of $$X$$ along any given meridian, and also that of $$Y$$ over the whole surface.

Let where the integration is performed along the given meridian from the pole to the parallel $$l$$, then Rh where the integration is performed along the parallel $$l$$ from the given meridian to the required point.

These methods imply that a complete magnetic survey of the earth's surface has been made, so that the values of $$X$$ or of $$Y$$ or of both are known for every point of the earth's surface at a given epoch. What we actually know are the magnetic components at a certain number of stations. In the civilized parts of the earth these stations are comparatively numerous; in other places there are large tracts of the earth's surface about which we have no data.

Magnetic Survey.

466.] Let us suppose that in a country of moderate size, whose greatest dimensions are a few hundred miles, observations of the declination and the horizontal force have been taken at a considerable number of stations distributed fairly over the country.

Within this district we may suppose the value of $$V$$ to be represented with sufficient accuracy by the formula Rh whence Rh

Let there be $$n$$ stations whose latitudes are $$l_1$$, $$l_2$$, ...&c. and longitudes $$\lambda_1$$, $$\lambda_2$$, &c., and let $$X$$ and $$Y$$ be found for each station.

Let