Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/438

$m(B+\beta )-n(A+\alpha )={\frac {D}{E}}\delta ,$

$m(B+\beta )-n(A'+\alpha )={\frac {D'}{E'}}\delta '.$

Hence

$n(A'-A)={\frac {D}{E}}\delta -{\frac {D'}{E'}}\delta '.$

If $\delta$ and $\delta '$, instead of being both apparently zero, had been only observed to be equal, then, unless we also could assert that $E=E'$, the right-hand side of the equation might not be zero. In fact, the method would be a mere modification of that already described.

The merit of the method consists in the fact that the thing observed is the absence of any deflexion, or in other words, the method is a Null method, one in which the non-existence of a force is asserted from an observation in which the force, if it had been different from zero by more than a certain small amount, would have produced an observable effect.

Null methods are of great value where they can be employed, but they can only be employed where we can cause two equal and opposite quantities of the same kind to enter into the experiment together.

In the case before us both $\delta$ and $\delta '$ are quantities too small to be observed, and therefore any change in the value of $E$ will not affect the accuracy of the result.

The actual degree of accuracy of this method might be ascertained by taking a number of observations in each of which $A'$ is separately adjusted, and comparing the result of each observation with the mean of the whole series.

But by putting $A'$ out of adjustment by a known quantity, as, for instance, by inserting at $A$ or at $B$ an additional resistance equal to a hundredth part of $A$ or of $B$, and then observing the resulting deviation of the galvanometer needle, we can estimate the number of degrees corresponding to an error of one per cent. To find the actual degree of precision we must estimate the smallest deflexion which could not escape observation, and compare it with the deflexion due to an error of one per cent.

^[1]If the comparison is to be made between $A$ and $B$, and if the positions of $A$ and $B$ are exchanged, then the second equation becomes

1. This investigation is taken from Weber's treatise on Galvanometry. Göttingen Transactions, x. p. 65.