Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/133

be formed. Of these one will be the surface of $A_{r}$ and its potential will be $p_{rr}$. If $A_{s}$ is placed in a hollow excavated in $A_{r}$ so as to be completely enclosed by it, then the potential of $A_{s}$ will also be $p_{rr}$.

If, however, $A_{s}$ is outside of $A_{r}$ its potential $p_{rs}$ will lie between $p_{rr}$ and zero.

For consider the lines of force issuing from the charged conductor $A_{r}$. The charge is measured by the excess of the number of lines which issue from it over those which terminate in it. Hence, if the conductor has no charge, the number of lines which enter the conductor must be equal to the number which issue from it. The lines which enter the conductor come from places of greater potential, and those which issue from it go to places of less potential. Hence the potential of an uncharged conductor must be intermediate between the highest and lowest potentials in the field, and therefore the highest and lowest potentials cannot belong to any of the uncharged bodies.

The highest potential must therefore be $p_{rr}$, that of the charged body $A_{r}$ , and the lowest must be that of space at an infinite distance, which is zero, and all the other potentials such as $p_{rs}$ must lie between $p_{rr}$ and zero.

If $A_{s}$ completely surrounds $A_{t}$ then $p_{rs}$ = $p_{rt}$.

91.] Theorem IV. None of the coefficients of induction are positive, and the sum of all those belonging to a single conductor is not numerically greater than the coefficient of capacity of that conductor, which is always positive.

For let $A_{r}$ be maintained at potential unity while all the other conductors are kept at potential zero, then the charge on $A_{r}$ is $q_{rr}$, and that on any other conductor $A_{s}$ is $q_{rs}$.

The number of lines of force which issue from $A_{r}$ is $p_{rr}$. Of these some terminate in the other conductors, and some may proceed to infinity, but no lines of force can pass between any of the other conductors or from them to infinity, because they are all at potential zero.

No line of force can issue from any of the other conductors such as $A_{s}$ , because no part of the field has a lower potential than $A_{s}$. If $A_{s}$ is completely cut off from $A_{r}$ by the closed surface of one of the conductors, then $q_{rs}$ is zero. If $A_{s}$ is not thus cut off, $q_{rs}$ is a negative quantity.

If one of the conductors $A_{t}$ completely surrounds $A_{r}$, then all the lines of force from $A_{r}$ fall on $A_{t}$ and the conductors within it,