Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/248

 $DA\frac{DA+OA}{2DA}=\frac{1}{2}\left(DA+OA\right),$

where $$O$$ is the centre of the circle of intersection.

In the same way the charge on the same segment due to the image at $$B$$ is $$\tfrac{1}{2}\left(DB+OB\right)$$, and so on, lines such as $$OB$$ measured from to the left being reckoned negative.

Hence the total charge on the segment whose centre is A is

$\frac{1}{2}(DA+DB+DC+etc.)+\frac{1}{2}(OA+OB+OC+etc.)$

$-\frac{1}{2}(DP+DQ+etc.)-\frac{1}{2}(OP+OQ+etc.)$

167.] The method of electrical images may be applied to any space bounded by plane or spherical surfaces all of which cut one another in angles which are submultiples of two right angles.

In order that such a system of spherical surfaces may exist, every solid angle of the figure must be trihedral, and two of its angles must be right angles, and the third either a right angle or a submultiple of two right angles.

Hence the cases in which the number of images is finite are–

(1) A single spherical surface or a plane.

(2) Two planes, a sphere and a plane, or two spheres intersecting at an angle $$\tfrac{\pi}{n}$$

(3) These two surfaces with a third, which may be either plane or spherical, cutting both orthogonally.

(4) These three surfaces with a fourth cutting the first two orthogonally and the third at an angle $$\tfrac{\pi}{n'}$$. Of these four surfaces one at least must be spherical.

We have already examined the first and second cases. In the first case we have a single image. In the second case we have $$2n-1$$ images arranged in two series in a circle which passes through the influencing point and is orthogonal to both surfaces. In the third case we have, besides these images, their images with respect to the third surface, that is, $$4n-1$$ images in all besides the influencing point.

In the fourth case we first draw through the influencing point a circle orthogonal to the first two surfaces, and determine on it the positions and magnitudes of the $$n$$ negative images and the $$n-1$$ positive images. Then through each of these 2$$n$$ points, including the influencing point, we draw a circle orthogonal to the third and fourth surfaces, and determine on it two series of