Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/253

 and the charge at the intersection of the four perpendiculars is

$-\frac{1}{\sqrt{\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}+\frac{1}{\delta^{2}}}}.$|undefined

System of Four Spheres Intersecting at Right Angles under the Action of an Electrified Point.

170.] Let the four spheres be $$A,B,C,D$$, and let the electrified point be $$O$$. Draw four spheres $$A_{1},B_{1},C_{1},D_{1}$$, of which any one, $$A_1$$, passes through and cuts three of the spheres, in this case $$B, C$$, and $$D$$, at right angles. Draw six spheres $$(ab), (ac), (ad), (be), (bd), (cd)$$, of which each passes through and through the circle of intersection of two of the original spheres.

The three spheres $$B_{1},C_{1},D_{1}$$ will intersect in another point besides $$O$$. Let this point be called $$A'$$, and let $$B', C'$$, and $$D'$$ be the intersections of $$C_{1},D_{1},A_{1}$$ of $$D_{1},A_{1},B_{1}$$, and of $$A_{1},B_{1},C_{1}$$ respectively. Any two of these spheres, $$A_{1},B_{1}$$, will intersect one of the six ($$cd$$) in a point ($$a'b'$$). There will be six such points.

Any one of the spheres, $$A_1$$, will intersect three of the six $$(ab), (ac), (ad)$$ in a point $$a'$$. There will be four such points. Finally, the six spheres $$(ab), (ac), (ad), (cd), (db), (bc)$$, will intersect in one point $$S$$.

If we now invert the system with respect to a sphere of radius $$R$$ and centre $$O$$, the four spheres $$A, B, C, D$$ will be inverted into spheres, and the other ten spheres will become planes. Of the points of intersection the first four $$A', B', C', D'$$ will become the centres of the spheres, and the others will correspond to the other eleven points in the preceding article. These fifteen points form the image of $$O$$ in the system of four spheres.

At the point $$A'$$, which is the image of $$O$$ in the sphere $$A$$, we must place a charge equal to the image of $$O$$, that is, $$-\tfrac{\alpha}{a}$$, where $$\alpha$$ is the radius of the sphere $$A$$, and $$a$$ is the distance of its centre from $$O$$. In the same way we must place the proper charges at $$B', C', D'$$.

The charges for each of the other eleven points may be found from the expressions in the last article by substituting $$\alpha',\beta',\gamma',\delta'$$ for $$\alpha,\beta,\gamma,\delta$$, and multiplying the result for each point by the distance of the point from $$O$$, where

$\alpha'=-\frac{\alpha}{a^{2}-\alpha^{2}},\ \beta'=-\frac{\beta}{b^{2}-\beta^{2}},\ \gamma'=-\frac{\gamma}{c^{2}-\gamma^{2}},\ \delta'=-\frac{\delta}{d^{2}-\delta^{2}}.$|undefined