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

281.] 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 $$\frac.$$ That of $$Q_p$$ in the expression for $$P_q$$ is $$ \frac. $$

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.

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 the part of the value of $$P_p$$ which depends on this electromotive force is and the part of the value of $$P_q$$ is

Therefore the coefficient of $$E_{rs}$$ in the value of $$P_p- P-q$$ is This is identical with the coefficient of $$E_{pq}$$ in the value of $$P_r-P_s.$$