Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/132

If we suppose that the probable errors of these equations are equal, as they will be if they depend on the determination of $D$ only, and if there is no uncertainty about $r$, then, by multiplying each equation by $r^{-3}$ and adding the results, we obtain one equation, and by multiplying each equation by $r^{-5}$ and adding we obtain another, according to the general rule in the theory of the combination of fallible measures when the probable error of each equation is supposed the same.

Let us write

$\sum {(Dr^{-3})}$ for $D_{1}r_{1}^{-3}+D_{2}r_{2}^{-3}+D_{3}r_{3}^{-3}+$ &c.,

and use similar expressions for the sums of other groups of symbols, then the two resultant equations may be written

$\sum (Dr^{-3})={\frac {2M}{H}}(\sum {(r^{-6})}+{A_{2}}\sum {(r^{-8})})$,

$\sum (Dr^{-5})={\frac {2M}{H}}(\sum {(r^{-8})}+{A_{2}}\sum {(r^{-10})})$,

whence

${\frac {2M}{H}}{\big \{}\sum (r^{-6})\sum (r^{-10})-{[\sum (r^{-8})]}^{2}{\big \}}=\sum (Dr^{-3})\sum (r^{-10})-\sum (Dr^{-5})\sum (r^{-8})$,

and

$A_{2}{\big \{}\sum (Dr^{-3})\sum (r^{-10})-\sum (Dr^{-5})\sum (r^{-8}){\big \}}$ $=\sum (Dr^{-5})\sum (r^{-6})-\sum (Dr^{-3})\sum (r^{-8})$

The value of $A_{2}$ derived from these equations ought to be less than half the square of the length of the magnet $M$. If it is not we may suspect some error in the observations. This method of observation and reduction was given by Gauss in the 'First Report of the Magnetic Association.'

When the observer can make only two series of experiments at distances $r_{1}$ and $r_{2}$, the value of ${\frac {2M}{H}}$ derived from these experiments is

$Q={\frac {2M}{H}}={\frac {D_{1}r_{1}^{5}-D_{2}r_{2}^{5}}{r_{1}^{2}-r_{2}^{2}}}$, \quad $A_{2}={\frac {D_{2}r_{2}^{3}-D_{1}r_{1}^{3}}{r_{1}^{2}-r_{2}^{2}}}r_{1}^{2}r_{2}^{2}$.

If $\delta D_{1}$ and $\delta D_{2}$ are the actual errors of the observed deflexions $D_{1}$ and $D_{2}$, the actual error of the calculated result $Q$ will be

$\delta Q={\frac {r_{1}^{5}\delta D_{1}-r_{2}^{5}\delta D_{2}}{r_{1}^{2}-r_{2}^{2}}}$.

If we suppose the errors $\delta D_{1}$ and $\delta D_{2}$ to be independent, and that the probable value of either is $\delta D$, then the probable value of the error in the calculated value of $Q$ will be $\delta Q$, where

$(\delta Q)^{2}={\frac {r_{1}^{10}+r_{2}^{10}}{(r_{1}^{2}-r_{2}^{2})^{2}}}(\delta D)^{2}$.