Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/276

and the potential at the point $(\rho ,\theta )$ is

${\begin{array}{l}\phi =E\log \left(e^{-2\rho _{0}}-2e^{-\rho _{0}}e^{\rho }\cos \theta +e^{2\rho }\right)\\\qquad -E\log \left(e^{2\rho _{0}}-2e^{-\rho _{0}}e^{\rho }\cos \theta +e^{2\rho }\right)+2E\rho _{0}\end{array}}$

(11)

This is the potential at the point $(\rho ,\theta )$ due to a charge $E$, placed at the point $(\rho _{0},0)$, with the condition that when $\rho =0,\ \phi =0$0.

In this case $\rho$ and $\theta$ are the conjugate functions in equations (5): $\rho$ is the logarithm of the ratio of the radius vector of a point to the radius of the circle, and $\theta$ is an angle.

The centre is the only singular point in this system of coordinates, and the line-integral of $\int {\tfrac {d\theta }{ds}}ds$ round a closed curve is zero or $2\pi$, according as the closed curve excludes or includes the centre.

EXAMPLE III. Neumann's Transformation of this Case^[1].

190.] Now let $\alpha$ and $\beta$ be any conjugate functions of $x$ and $y$, such that the curves ($\alpha$) are equipotential curves, and the curves ($\beta$) are lines of force due to a system consisting of a charge of half a unit at the origin, and an electrified system disposed in any manner at a certain distance from the origin.

Let us suppose that the curve for which the potential is a is a closed curve, such that no part of the electrified system except the half-unit at the origin lies within this curve.

Then all the curves ($\alpha$) between this curve and the origin will be closed curves surrounding the origin, and all the curves ($\beta$) will meet in the origin, and will cut the curves ($\alpha$) orthogonally.

The coordinates of any point within the curve ($\alpha _{0}$) will be determined by the values of $\alpha$ and $\beta$ at that point, and if the point travels round one of the curves $\alpha$ in the positive direction, the value of $\beta$ will increase by $2\pi$ for each complete circuit.

If we now suppose the curve ($\alpha _{0}$) to be the section of the inner surface of a hollow cylinder of any form maintained at potential zero under the influence of a charge of linear density $E$ on a line of which the origin is the projection, then we may leave the external electrified system out of consideration, and we have for the potential at any point ($\alpha$) within the curve

$\phi =2E\left(\alpha -\alpha _{0}\right)$

(12)

and for the quantity of electricity on any part of the curve $\alpha _{0}$ between the points corresponding to $\beta _{1}$ and $\beta _{2}$,

$Q=2E\left(\beta _{1}-\beta _{2}\right)$

(13)

1. See Crelle's Journal, 1861.