Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/191

a corresponding series of values of $\theta$, and if $E$ be an integer, the number of corresponding lines of force, including the axis, will be equal to $E$.

We have therefore a method of drawing lines of force so that the charge of any centre is indicated by the number of lines which converge to it, and the induction through any surface cut off in the way described is measured by the number of lines of force which pass through it. The dotted straight lines on the left hand side of Fig. 6 represent the lines of force due to each of two electrified points whose charges are 10 and -10 respectively.

If there are two centres of force on the axis of the figure we may draw the lines of force for each axis corresponding to values of $\Psi _{1}$ and $\Psi _{2}$, and then, by drawing lines through the consecutive intersections of these lines, for which the value of $\Psi _{1}+\Psi _{2}$ is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The continuous curves on the left hand side of Fig. 6 represent the lines of force due to the two electrified points acting at once.

After the equipotential surfaces and lines of force have been constructed by this method the accuracy of the drawing may be tested by observing whether the two systems of lines are every where orthogonal, and whether the distance between consecutive equipotential surfaces is to the distance between consecutive lines of force as half the distance from the axis is to the assumed unit of length.

In the case of any such system of finite dimensions the line of force whose index number is $\Psi$ has an asymptote which passes through the centre of gravity of the system, and is inclined to the axis at an angle whose cosine is $1-2{\frac {\Psi }{E}}$, where $E$ is the total electrification of the system, provided $\Psi$ is less than $E$. Lines of force whose index is greater than $E$ are finite lines.

The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series.

The theory of equipotential surfaces and lines of force in two dimensions will be given when we come to the theory of conjugate functions^[1].

1. See a paper ‚On the Flow of Electricity in Conducting Surfaces‘, by Prof. W. R. Smith, Proc. R. S. Edin., 1869-70, p. 79.