156] ELECTRICAL IMAGES. 301 



It is to be noticed that if the original distribution is an equi- 

 potential one, which is the case if it is on the surface of a 

 conductor, the image will not be equipotential, on account of the 

 variable factor l/l', but if we place at 0, the center of inversion, a 

 charge R V, its potential R V/l' added to the potential due to 

 the image gives zero, an equipotential distribution. Consequently 

 any problem of electrical equilibrium whose solution is known 

 gives us the solution of the problem of induction by any point- 

 charge on a conductor whose surface is the inverse of the given 

 conductor with respect to the point at which the inducing charge 

 is placed. Conversely the solution of a problem of induction by 

 a point-charge gives us the solution of a problem of undisturbed 

 equilibrium. Thus the solutions of the problems treated above 

 furnish us new solutions. 



The image of a sphere is a sphere (including a plane as a 

 special case) for the equation of a sphere 



A (aj+ 2/ 2 + *) + Bx+ Cy + Dz + E = 0, 



becomes, using the equations (3), 



' 



*'*' 



that is, 



E (x* + y' 2 + z'*) + BB?x' + CRy + D&J + AR*=Q. 



If A is zero, we have originally a plane, which inverts into a 

 sphere passing through the origin, while if E is zero, the sphere 

 through the origin inverts into a plane. 



As an example of the method, let us invert a sphere of radius 

 a/2 charged to potential V about a point on its surface, with 

 radius of inversion, R = a. The sphere inverts into a plane 

 tangent to the sphere at the point diametrically opposite the 

 center of inversion. The charge of the sphere being Fa/2, the 

 surface density is 



F 

 ~ 



Consequently 



,a 3 Fa 8 



