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

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.

where the integration is performed along the given meridian from the pole to the parallel $l$, then

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

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.