220-222] Images 193 



various fields, however, continually diminish, so that although we get an 

 infinite series to express the potential, this series is convergent. As we shall 

 see, this series can be summed as a definite integral, or it may be that a good 

 approximation will be obtained by taking only a finite number of terms. 



The total charge on A is clearly the sum of the original charge Fa plus 

 the strengths of the images A', A", ... etc., for this sum measures the 

 aggregate strength of the tubes of force which end on A. Similarly the 

 charge on B is the sum of the strengths of the images at B f , B", .... 



To obtain the field corresponding to given potentials of both A and B we 

 superpose on to the field already found, the similar field obtained by raising 

 B to the required potential while that of A remains zero. 



If #11, #22, #12 are the coefficients of capacity and induction, the total charge 

 on A when B is to earth and F= 1 is # n ; similarly that on B is # 12 . In this 

 way we can find the coefficients # n , # 12 from the series of images already 

 obtained. The result is found to be 



a 2 6 a 3 6 2 



db a 2 6 2 



and from symmetry 



As far as , these results clearly agree with those of 116. 

 c 



222. The series for q n , 5' 12 , ^22 have been put in a more manageable form by Poisson 

 and Kirchhoff. 



Let A a denote the position of the sth of the series of points A', A", ..., and B g the sth 

 of the series ', ", ... ; then A 8 is the image of B 8 in the sphere of radius a, and similarly 

 B s is the image of A 8 _i in the sphere of radius 6. Let a g =AA 8 , b 8 = BB 8 , and let the 

 charges at J a , B 8 be e s , e' 8 respectively. 



Then a g (c b 8 ) = a 2 since A 8 is the image of JB 8 , 



&a(c-a 8 _i) = & 2 B* A-i- 



Further, by comparing the strengths of a charge and its image, 



80 that -- 



and similarly e' 8 = - - - =- - - e' 8 _ l . 



(c-a g _ 1 )(c-b 8 _ 1 ) 



,, T , ,i /. 

 We have therefore 



- - - -- - - 

 e-i (c-bJ^-ag.J a b ab 



and e * -~ ~ 



ab ab a' 



13 



