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

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

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.

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

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