Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/150

is $2\pi$, and in order to define which of these is to be taken in any particular case we must make some restriction as to the line along which we are to integrate the force from the point where $V=0$ to the required point.

In this case the region in which the condition of having a potential is fulfilled is the cyclic region surrounding the axis of z, this axis being a line in which the forces are infinite and therefore not itself included in the region.

The part of the infinite plane of $xz$ for which $x$ is positive may be taken as a diaphragm of this cyclic region. If we begin at a point close to the positive side of this diaphragm, and integrate along a line which is restricted from passing through the diaphragm, the line-integral will be restricted to that value of $V$ which is positive but less than $2\pi$.

Let us now suppose that the region bounded by the closed surface $S$ in Green s Theorem is a cyclic region of any number of cycles, and that the function $V$ is a many-valued function having any number of cyclic constants.

The quantities ${\frac {dV}{dx}}$, ${\frac {dV}{dy}}$, and ${\frac {dV}{dz}}$ will have definite values at all points within $S$, so that the volume-integral

$\iiint K\left({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}}\right)$

has a definite value, $\sigma$ and $p$ have also definite values, so that if $U$ is a single valued function, the expression

$\iint \sigma UdS+\iiint \rho Udxdydz$

has also a definite value.

The expression involving $V$ has no definite value as it stands, for $V$ is a many-valued function, and any expression containing it is many-valued unless some rule be given whereby we are directed to select one of the many values of V at each point of the region.

To make the value of $V$ definite in a region of $n$ cycles, we must conceive $n$ diaphragms or surfaces, each of which completely shuts one of the channels of communication between the parts of the cyclic region. Each of these diaphragms reduces the number of cycles by unity, and when n of them are drawn the region is still a connected region but acyclic, so that we can pass from any one point to any other without cutting a surface, but only by reconcileable paths.