Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/243

and has therefore a constant potential $P$, then in the transformed system the image of the surface will have a potential $P{\tfrac {R}{r'}}$. But by placing at $O$, the centre of inversion, a quantity of electricity equal to $-PR$, the potential of the transformed surface is reduced to zero.

Hence, if we know the distribution of electricity on a conductor when insulated in open space and charged to the potential $P$, we can find by inversion the distribution on a conductor whose form is the image of the first under the influence of an electrified point with a charge $-PR$ placed at the centre of inversion, the conductor being in connexion with the earth.

163.] The following geometrical theorems are useful in studying cases of inversion.

Every sphere becomes, when inverted, another sphere, unless it passes through the centre of inversion, in which case it becomes a plane.

If the distances of the centres of the spheres from the centre of inversion are $a$ and $a'$, and if their radii are $\alpha$ and $\alpha '$, and if we define the power of the sphere with respect to the centre of in version to be the product of the segments cut off by the sphere from a line through the centre of inversion, then the power of the first sphere is $a^{2}-\alpha ^{2}$, and that of the second is $a'^{2}-\alpha '^{2}$. We have in this case

${\frac {a'}{a}}={\frac {\alpha '}{\alpha }}={\frac {R^{2}}{a^{2}-\alpha ^{2}}}={\frac {a'^{2}-\alpha '^{2}}{R^{2}}},$

(19)

or the ratio of the distances of the centres of the first and second spheres is equal to the ratio of their radii, and to the ratio of the power of the sphere of inversion to the power of the first sphere, or of the power of the second sphere to the power of the sphere of inversion.

The centre of either sphere corresponds to the inverse point of the other with respect to the centre of inversion.

In the case in which the inverse surfaces are a plane and a sphere, the perpendicular from the centre of inversion on the plane is to the radius of inversion as this radius is to the diameter of the sphere, and the sphere has its centre on this perpendicular and passes through the centre of inversion.

Every circle is inverted into another circle unless it passes through the centre of inversion, in which case it becomes a straight line.