Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/132

This function is known by the name of Green's Function.

The coefficients of induction $q_{rs}$ and $q_{sr}$ are also equal. This is easily seen from the process by which these coefficients are obtained from the coefficients of potential. For, in the expression for $q_{rs}$, $p_{rs}$ and $p_{sr}$ enter in the same way as $p_{sr}$ and $p_{rs}$ do in the expression for $q_{sr}$ . Hence if all pairs of coefficients $p_{rs}$ and $p_{sr}$ are equal, the pairs $q_{rs}$ and $q_{sr}$ are also equal.

89.] Theorem II. Let a charge $E_{r}$ be placed on $A_{r}$, and let all the other conductors he at potential zero, and let the charge induced on $A_{s}$ be $-n_{rs}E_{r}$, then if $A_{r}$ is discharged and insulated, and $A_{s}$ brought to potential $V_{s}$, the other conductors being at potential zero, then the potential of $A_{r}$ will be $+n_{rs}V_{s}$.

For, in the first case, if $V_{r}$ is the potential of $A_{r}$, we find by equations (2),

$E_{s}\,=\,q_{rs}\,V_{r}{\color {White}xxxx}$, and ${\color {White}xxx}E_{r}=q_{rr}V_{r}$.

Hence ${\color {White}xxxx}E_{s}={\frac {q_{rs}}{q_{rr}}}E_{r}{\color {White}xxxx}$, and ${\color {White}xxxx}n_{rs}=-{\frac {q_{rs}}{q_{rr}}}$

In the second case, we have

$E_{r}=0=q_{rr}V_{r}+q_{rs}V_{s}\,\!$.

Hence ${\color {White}xxxxx}V_{r}=-{\frac {q_{rs}}{q_{rr}}}V_{s}=n_{rs}V_{s}$.

From this follows the important theorem, due to Green: If a charge unity, placed on the conductor $A_{0}$ in presence of conductors $A_{1}$, $A_{2}$, &c. at potential zero induces charges $-n_{1}$, $-n_{2}$, &c. in these conductors, then, if $A_{0}$ is discharged and insulated, and these conductors are maintained at potentials $V_{1}$, $V_{2}$, &c., the potential of $A_{0}$ will be

${n_{1}}{V_{1}}+{n_{2}}{V_{2}}+$&c.

The quantities $(n)$ are evidently numerical quantities, or ratios.

The conductor $A_{0}$ may be supposed reduced to a point, and $A_{1}$, $A_{2}$, &c. need not be insulated from each other, but may be different elementary portions of the surface of the same conductor. We shall see the application of this principle when we investigate Green's Functions.

90.] Theorem III. The coefficients of potential are all positive,but none of the coefficients $p_{rs}$ is greater than $p_{rr}$ or $p_{ss}$.

For let a charge unity be communicated to $A_{r}$, the other conductors being uncharged. A system of equipotential surfaces will