Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/270

the surface-integral of a surface whose projection on the plane of $xy$ is the curve $AB$ will be $k\left(\beta _{B}-\beta _{A}\right)$, where $\beta _{A}$ and $\beta _{B}$ are the values of $\beta$ at the extremities of the curve.

If a series of curves corresponding to values of $\alpha$ in arithmetical progression is drawn on the plane, and another series corresponding to a series of values of $\beta$ having the same common difference, then the two series of curves will everywhere intersect at right angles, and, if the common difference is small enough, the elements into which the plane is divided will be ultimately little squares, whose sides, in different parts of the field, are in different directions and of different magnitude, being inversely proportional to $R$.

If two or more of the equipotential lines ($\alpha$) are closed curves enclosing a continuous space between them, we may take these for the surfaces of conductors at potentials $\left(V_{0}+k\alpha _{1}\right)$, $\left(V_{0}+k\alpha _{2}\right)$, &c. respectively. The quantity of electricity upon any one of these between the lines of force $\beta _{1}$ and $\beta _{2}$ will be ${\tfrac {k}{4\pi }}\left(\beta _{2}-\beta _{1}\right)$.

The number of equipotential lines between two conductors will therefore indicate their difference of potential, and the number of lines of force which emerge from a conductor will indicate the quantity of electricity upon it.

We must next state some of the most important theorems relating to conjugate functions, and in proving them we may use either the equations (1), containing the differential coefficients, or the original definition, which makes use of imaginary symbols.

184.] THEOREM I. If $x'$ and $y'$ are conjugate functions with respect to $x$ and $y$, and if $x''$ and $y''$ are also conjugate functions with respect to $x$ and $y$, then the functions $x'+x''$ and $y'+y''$ will be conjugate functions with respect to $x$ and $y$.

For

${\frac {dx'}{dx}}={\frac {dy'}{dy}}$ and ${\frac {dx''}{dx}}={\frac {dy''}{dy}}$;

therefore

${\frac {d(x'+x'')}{dx}}={\frac {d(y'+y'')}{dy}}$.

Also

${\frac {dx'}{dy}}=-{\frac {dy'}{dx}}$ and ${\frac {dx''}{dy}}=-{\frac {dy''}{dx}}$;

therefore

${\frac {d(x'+x'')}{dy}}=-{\frac {d(y'+y'')}{dx}}$;

or $x+x''$ and $y'+y''$ are conjugate with respect to $x$ and $y$.