298 ELECTROSTATICS. [PT. II. CH. VII. 



value, we have a repulsion, when 



or \E\> 



ear 



156. Electrical Images in a Sphere. Points which are 

 electrical images of each other, besides having the properties 

 connected with equipotential layers described above, possess 

 peculiar reciprocal properties with respect to the portions of space 

 in which they are respectively situated. There thus arises a 

 method of finding from the known solutions of electrostatic 

 problems a new class of problems whose solutions can be found. 

 This method of electrical images was discovered by Lord Kelvin in 

 1848*. Suppose as before that A and B are inverse points with 

 reference to the sphere of radius R, A being outside. Let M and 

 M 1 (M outside) be two other inverse points situated at distances 

 I and I' from the center, and at distances r and r respectively 

 from A and B. Then the triangles 0AM and OM'B are similar, 

 since ab II' = R*. Suppose a charge e placed at A, and a charge 

 e = eR/a placed at B. If we call V the potential at M due to 

 the charge e, and F' the potential at M' due to the charge e', 

 we have 



V _e r_J r__Rl_ _l^_ _R 

 T~~r'' ~e~ e ' r'~ a b~ R~ I' ' 



If then we have any number of electrified points such as A, 

 and find their images B, and if V be the potential of the system 

 A at any external point, M, then 



RV IV 



& r- -* 



will be the potential at M ' the inverse point to M, of the system 

 B which is the electrical image of the system A. 



We shall give an analytical proof of the same proposition, 

 based on the method of curvilinear coordinates. If x, y, z are the 

 coordinates of the point M, of, y' y /, those of the point M', we have 

 x jx y'\y z'jz and since IV = R 2 , 



-_ 



'~W~* y 



z I' 2 = a/ 2 + 2 4- 



* Papers on Electrostatics and Magnetism, p. 144. 



