Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/377

say $A_{q},$ over that of $A_{n}$ may be determined. We may then determine the current between $A_{p}$ and $A_{q}$ from equation (1), and so solve the problem completely.

281.] We shall now demonstrate a reciprocal property of any two conductors of the system, answering to the reciprocal property we have already demonstrated for statical electricity in Art. 88.

The coefficient of $Q_{q}$ in the expression for $P_{p}$ is ${\frac {D_{pq}}{D}}.$ That of $Q_{p}$ in the expression for $P_{q}$ is ${\frac {D_{qp}}{D}}.$

Now $D_{pq}$ differs from $D_{qp}$ only by the substitution of the symbols such as $K_{qp}$ for $K_{pq}$. But, by equation (2), these two symbols are equal, since the conductivity of a conductor is the same both ways.

Hence

$D_{pq}=D_{qp}.$

(11)

It follows from this that the part of the potential at $A_{p}$ arising from the introduction of a unit current at $A_{q}$ is equal to the part of the potential at $A_{q}$ arising from the introduction of a unit current at $A_{p}.$

We may deduce from this a proposition of a more practical form.

Let $A,\,B,\,C,\,D$ be any four points of the system, and let the effect of a current $Q,$ made to enter the system at $A$ and leave it at $B,$ be to make the potential at $C$ exceed that at $D$ by $P$. Then, if an equal current $Q$ be made to enter the system at $C$ and leave it at $D,$ the potential at $A$ will exceed that at $B$ by the same quantity $P$.

We may also establish a property of a similar kind relating to the effect of the internal electromotive force $E_{rs},$ acting along the conductor which joins the points $A_{r}$ and $A_{s}$ in producing an external electromotive force on the conductor from $A_{p}$ to $A_{q},$ that is to say, a difference of potentials $P_{p}-P_{q}.$ For since

$E_{rs}=-E_{sr},$

the part of the value of $P_{p}$ which depends on this electromotive force is

${\frac {1}{D}}(D_{pr}-D_{ps})E_{rs},$

and the part of the value of $P_{q}$ is

${\frac {1}{D}}(D_{qr}-D_{qs})E_{rs}.$

Therefore the coefficient of $E_{rs}$ in the value of $P_{p}-P-q$ is

${\frac {1}{D}}{D_{pr}+D+{qs}-D_{ps}-D_{qr}}.$

(12)

This is identical with the coefficient of $E_{pq}$ in the value of $P_{r}-P_{s}.$