Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/154

$l_{0}$ and $\lambda _{0}$ may be called the latitude and longitude of the central station. Let

then $X_{0}$ and $Y_{0}$ are the values of $X$ and $Y$ at the imaginary central station, then

We have $n$ equations of the form of (11) and $n$ of the form (12). If we denote the probable error in the determination of $X$ by $\xi$, and that of $Y\cos l$ by $\eta$, then we may calculate $\xi$ and $\eta$ on the supposition that they arise from errors of observation of $H$ and $\delta$.

Let the probable error of $H$ be $h$, and that of $\delta$, $d$, then since

If the variations of $X$ and $Y$ from their values as given by equations of the form (11) and (12) considerably exceed the probable errors of observation, we may conclude that they are due to local attractions, and then we have no reason to give the ratio of $\xi$ to $\eta$ any other value than unity.

According to the method of least squares we multiply the equations of the form (11) by $\eta$, and those of the form (12) by $\xi$ to make their probable error the same. We then multiply each equation by the coefficient of one of the unknown quantities $B_{1}$, $B_{2}$, or $B_{3}$ and add the results, thus obtaining three equations from which to find $B_{1}$, $B_{2}$, and $B_{3}$.

in which we write for conciseness,

By calculating $B_{1}$, $B_{2}$, and $B_{3}$, and substituting in equations (11) and (12), we can obtain the values of $X$ and $Y$ at any point within the limits of the survey free from the local disturbances