294 ELECTROSTATICS. [PT. II. CH. VII. 



solves Dirichlet's problem for the left-hand side of the plane. If 

 we suppose a charge of a positive unit placed at A, and a negative 

 unit at B, the plane of symmetry will be an equipotential surface 

 of zero potential, and we may apply the theorem of equipotential 

 layers. If the plane of symmetry is made conducting, and the 

 charge B removed, the conducting plane receives a charge 1 

 which screens the space on the right from the action of A. The 

 surface density on the plane is O G = cos 0/2-Trr 2 , so that the 

 whole charge on the plane is, applying Gauss's theorem, 

 1 /7cos0 , 



This is an example of the second theorem of 150, the space 

 on the right being considered internal. 



The charge 1 at B is said to be the electrical image in the 

 plane of the charge + 1 at A. 



Two point-charges A and B are said to be electrical images of 

 each other in a certain closed surface separating them if either one, 

 gay B, produces in the portion of space in which the other, A, lies, 

 ;he same effect as would be produced there by the charge induced 

 n the surface made conducting and connected to earth, by the 

 joint A alone, the image B, being removed. 



153. Planes intersecting in a sub-multiple of two right 

 angles. 



Let us seek Green's function for a portion of space lying in the 

 acute angle between two planes intersecting in an angle which is 

 equal to two right angles divided by an integer. Let the planes 

 be denoted by 1 and 2, let the pole be P, and let P l be the 

 geometrical image of P in 1, P 2 that of P x in 2, P 3 that of P 2 in 1, 

 and so on alternately in the two planes. Let Qi be the image of 

 P in 2, Q z that of Q l in 1, Q 3 that of Q 2 in 2, and so on. Since the 

 angle is a submultiple of TT it is easily seen that the series of 

 images will be finite, the Q's and P's finally coinciding. Let the 

 distance of any point from P be denoted by r, from any P s by r s , 

 and from any Q s by r s f . Then the reciprocal of any distance r s or 

 r s ' is a harmonic function in the space between the planes since 

 none of the images lie in that space. Also for all points lying on 

 the plane 1, 



i_I = o l-l=o I-i=o l-l=o 



r r, r,' r,' r, r z r,' r t ' 



