Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/131

88.] 88.]. The coefficients of $$A_r$$ relative to $$A_8$$ are equal to those of $$A_8$$ relative to $$A_r$$ .

If $$E_r$$, the charge on $$A_r$$, is increased by $$\delta E_r$$, the work spent in bringing $$\delta E_r$$ from an infinite distance to the conductor $$A_r$$ whose  potential is $$V_r$$, is by the definition of potential in Art. 70,

and this expresses the increment of the electric energy caused by this increment of charge.

If the charges of the different conductors are increased by $$\delta E_1$$, &c., the increment of the electric energy of the system will be

If, therefore, the electric energy $$Q$$ is expressed as a function of the charges $$E_1$$, $$E_2$$, &c., the potential of any conductor may be  expressed as the partial differential coefficient of this function with  respect to the charge on that conductor, or

Since the potentials are linear functions of the charges, the energy must be a quadratic function of the charges. If we put

for the term in the expansion of $$Q$$ which involves the product $$E_r E_s$$, then, by differentiating with respect to $$E_s$$, we find the term  of the expansion of $$V_s$$ which involves $$E_r$$ to be $$CE_r$$.

Differentiating with respect to $$E_r$$, we find the term in the expansion of $$V_r$$ which involves $$E_s$$ to be $$CE_s$$.

Comparing these results with equations (1), Art. 86, we find or, interpreting the symbols $$p_{rs}$$ and $$p_{sr}$$ :—

The potential of $$A_s$$ due to a unit charge on $$A_r$$ is equal to the potential of $$A_r$$ due to a unit charge on $$A_s$$.

This reciprocal property of the electrical action of one conductor on another was established by Helmholtz and Sir W. Thomson.

If we suppose the conductors $$A_r$$ and $$A_s$$ to be indefinitely small, we have the following reciprocal property of any two points :

The potential at any point $$A_s$$, due to unit of electricity placed at $$A_r$$ in presence of any system of conductors, is a function of the  positions of $$A_r$$ and $$A_s$$ in which the coordinates of $$A_r$$ and of $$A_s$$  enter in the same manner, so that the value of the function is  unchanged if we exchange $$A_r$$ and $$A_s$$.