Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/454

If the deflexion of the galvanometer remains unaltered, we know that $OB$ is conjugate to $CA$, whence $c\gamma =a\alpha$, and $a$, the resistance of the battery, is obtained in terms of known resistances $c,\,\gamma ,\,\alpha$.

When the condition $c\gamma =a\alpha$ is fulfilled, then the current through the galvanometer is

$y={\frac {E\alpha }{b\alpha +C(b+\alpha +\gamma )}},$

and this is independent of the resistance $\beta$ between $O$ and $B$. To test the sensibility of the method let us suppose that the condition $c\gamma =a\alpha$ is nearly, but not accurately, fulfilled, and that $y_{0}$ is the

Fig. 37.

current through the galvanometer when $O$ and $B$ are connected by a conductor of no sensible resistance, and $y_{1}$ the current when $O$ and $B$ are completely disconnected.

To find these values we must make $\beta$ equal to $0$ and to $\infty$ in the general formula for $y$, and compare the results.

In this way we find

${\frac {y_{0}-y_{1}}{y}}={\frac {\alpha }{\gamma }}{\frac {c\gamma -a\alpha }{(c+\alpha )(\alpha +\gamma )}}$,

where $y_{0}$ and $y_{0}$ are supposed to be so nearly equal that we may, when their difference is not in question, put either of them equal to $y$, the value of the current when the adjustment is perfect.

The resistance, $c$, of the conductor $AB$ should be equal to $a$, that of the battery, $\alpha$ and $\gamma$, should be equal and as small as possible, and $b$ should be equal to $a+y$.

Since a galvanometer is most sensitive when its deflexion is small, we should bring the needle nearly to zero by means of fixed magnets before making contact between $O$ and $B$.

In this method of measuring the resistance of the battery, the current in the battery is not in any way interfered with during the operation, so that we may ascertain its resistance for any given