Page:A Dynamical Theory of the Electromagnetic Field.pdf/19

Then calling the quantity

${\frac {p}{P}}-{\frac {q}{Q}}-{\frac {r}{R}}+{\frac {s}{S}}+h\left({\frac {1}{P}}-{\frac {1}{Q}}\right)+k\left({\frac {1}{R}}-{\frac {1}{P}}\right)+l\left({\frac {1}{R}}+{\frac {1}{Q}}\right)-m\left({\frac {1}{P}}+{\frac {1}{S}}\right)+n\left({\frac {1}{Q}}-{\frac {1}{S}}\right)+o\left({\frac {1}{S}}-{\frac {1}{R}}\right)=\tau$

(26)

we find

${\frac {Z}{z}}={\frac {2\sin {\frac {1}{2}}\alpha }{\tan \theta }}{\frac {T}{\pi }}={\frac {\tau }{1-\rho }}$

(27)

In determining $\tau$ by experiment, it is best to make the alteration of resistance in one of the arms by means of the arrangement described by Mr. Jenkin in the Report of the British Association for 1863, by which any value of $\rho$ from 1 to 1.01 can be accurately measured.

We observe ($\alpha$) the greatest deflection due to the impulse of induction when the galvanometer is in circuit, when the connections are made, and when the resistances are so adjusted as to give no permanent current.

We then observe ($\beta$) the greatest deflection produced by the permanent current when the resistance of one of the arms is increased in the ratio of 1 to $\rho$, the galvanometer not being in circuit till a little while after the connection is made with the battery.

In order to eliminate the effects of resistance of the air, it is best to vary $\rho$ till $\beta =2\alpha$ nearly; then

$\tau =T{\frac {1}{\pi }}(1-\rho ){\frac {2\sin {\frac {1}{2}}\alpha }{\tan \theta \beta }}$

(28)

If all the arms of the balance except $P$ consist of resistance coils of very fine wire of no great length and doubled before being coiled, the induction coefficients belonging to these coils will be insensible, and $\tau$ will be reduced to ${\tfrac {p}{P}}$. The electric balance therefore affords the means of measuring the self-induction of any circuit whose resistance is known.

(46) It may also be used to determine the coefficient of induction between two circuits, as for instance, that between P and S which we have called $m$; but it would be more convenient to measure this by directly measuring the current, as in (37), without using the balance. We may also ascertain the equality of ${\tfrac {p}{P}}$ and ${\tfrac {q}{Q}}$ by there being no current of induction, and thus, when we know the value of $p$, we may determine that of $q$ by a more perfect method than the comparison of deflections.

Exploration of the Electromagnetic Field.

(47) Let us now suppose the primary circuit $A$ to be of invariable form, and let us explore the electromagnetic field by means of the secondary circuit $B$, which we shall suppose to be variable in form and position.

We may begin by supposing $B$ to consist of a short straight conductor with its extremities sliding on two parallel conducting rails, which are put in connection at some distance from the sliding-piece.