THREE DIELECTRICS SEPARATED BY PARALLEL PLANES. 133 



_. . _ :_V^ .. . . . . . . _. .,__-..-,-, IKJm _ 



The density at every point is equal to the algebraical sum of the 

 densities of all the layers superposed. 



The potential V, at a point P in the former medium, may be con- 

 sidered as produced by the mass m, and all the images situate at B, 

 B lf B 2 ...,etc. 



At the different points B I} B 2 . . . , there are two different images 

 arising from the layers on the two planes Q and Q', and we have : 



atBj 



m' - m'y = m'(i -y) = m(i- y 2 )/, 

 atB 2 



' = -m'yy'(i-y) = - m(i -y 2 )/.yy', 



at B n+1 



m'(yy'Ym'(yy'Yy= m'(yy') n (* -?)= 

 which gives 



The potential V 3 in the third medium is produced, in like manner, 

 by the images situate at points A, A 1 A 2 . . . , on which the masses 

 are : 



m + my + m' = m(i + y) (i + y'), 

 - m'y - m'yy = - m (i + y) (i + /) yy', 



(yy-i + m'y-Y n ) =m(i + y)(i + y) (yy'Y ; 



we have then 



Lastly, in the second medium, comprised between the planes Q 

 and Q', the potential is due to the mass m, to the images at A v A 2 . . . 

 of the layers of the plane Q, and to the images at B lf B 2 . . . of the 

 layers of the plane Q'. We shall find in like manner 



r_L_jzi 



|_PA PA X 



77' (rrT 



h ~PA7" 

 yy' (yy') 2 



PB 2 PB 3 



If the third medium is identical with the first, we simply put 

 then we get y'= y. 



