Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/246

For the alternate images $P_{1},P_{2},P_{3}$ are ranged round the circle at angular intervals equal to $2AOB$, and the intermediate images $Q_{1},Q_{2},Q_{3}$ are at intervals of the same magnitude. Hence, if $2AOB$ is a sub multiple of 2$\pi$, there will be a finite number of images, and none of these will fall within the angle $AOB$. If, however, $AOB$ is not a submultiple of it, it will be impossible to represent the actual electrification as the result of a finite series of electrified points.

If $AOB={\frac {\pi }{n}}$, there will be $n$ negative images $Q_{1},Q_{2},$ &c., each equal and of opposite sign to $P$, and $n-1$ positive images $P_{2},P_{3}$, &c., each equal to $P$, and of the same sign.

The angle between successive images of the same sign is ${\tfrac {2\pi }{n}}$. If we consider either of the conducting planes as a plane of symmetry, we shall find the positive and negative images placed symmetrically with regard to that plane, so that for every positive image there is a negative image in the same normal, and at an equal distance on the opposite side of the plane.

If we now invert this system with respect to any point, the two planes become two spheres, or a sphere and a plane intersecting at an angle ${\tfrac {\pi }{n}}$, the influencing point $P$ being within this angle.

The successive images lie on the circle which passes through $P$ and intersects both spheres at right angles.

To find the position of the images we may either make use of the principle that a point and its image are in the same radius of the sphere, and draw successive chords of the circle beginning at $P$ and passing through the centres of the two spheres alternately.

To find the charge which must be attributed to each image, take any point in the circle of intersection, then the charge of each image is proportional to its distance from this point, and its sign is positive or negative according as it belongs to the first or the second system.

166.] We have thus found the distribution of the images when any space bounded by a conductor consisting of two spherical surfaces meeting at an angle ${\tfrac {\pi }{n}}$, and kept at potential zero, is influenced by an electrified point.

We may by inversion deduce the case of a conductor consisting