Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/130

then we shall have the following equations determining the potentials in terms of the charges:

${\begin{matrix}V_{1}=p_{11}E_{1}\ldots +p_{r1}E_{r}\ldots +p_{n1}E_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\V_{s}=p_{1s}E_{1}\ldots +p_{rs}E_{r}\ldots +p_{ns}E_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\V_{n}=p_{1n}E_{1}\ldots +p_{rn}E_{r}\ldots +p_{nn}E_{n},\end{matrix}}$

(1)

We have here $n$ linear equations containing $n^{2}$ coefficients of potential.

87.] By solving these equations for $E_{1}$, $E_{2}$ , &c. we should obtain $n$ equations of the form

${\begin{matrix}E_{1}=q_{11}V_{1}\ldots +q_{1s}V_{s}\ldots +q_{1n}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{r}=q_{r1}V_{1}\ldots +q_{rs}V_{s}\ldots +q_{rn}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{n}=q_{n1}V_{1}\ldots +q_{ns}V_{s}\ldots +q_{nn}V_{n},\end{matrix}}$

(2)

The coefficients in these equations may be obtained directly from those in the former equations. They may be called Coefficients of Induction.

Of these $q_{11}$ is numerically equal to the quantity of electricity on $A_{l}$ when $A_{l}$ is at potential unity and all the other bodies are at potential zero. This is called the Capacity of $A_{1}$. It depends on the form and position of all the conductors in the system.

Of the rest $q_{rs}$ is the charge induced on $A_{r}$ when $A_{s}$ is maintained at potential unity and all the other conductors at potential zero. This is called the Coefficient of Induction of $A_{s}$ on $A_{r}$.

The mathematical determination of the coefficients of potential and of capacity from the known forms and positions of the conductors is in general difficult. We shall afterwards prove that they have always determinate values, and we shall determine their values in certain special cases. For the present, however, we may suppose them to be determined by actual experiment.

Dimensions of these Coefficients.

Since the potential of an electrified point at a distance $r$ is the charge of electricity divided by the distance, the ratio of a quantity of electricity to a potential may be represented by a line. Hence all the coefficients of capacity and induction $(q)$ are of the nature of lines, and the coefficients of potential $(p)$ are of the nature of the reciprocals of lines.