W n* Electricity in Equilibrium, -^^^'i'^ '^^^ ^ 57 



nitely near the former points are, according to the law stated 



Q 01 



above, y- and -r, it follows that the intensities of the imagined 



distributions on S and S', in the nieghbourhood of the points 

 considered, are 



Hence, if U, U' be the potentials at S, S', due to A and B alone, 

 and V the potential at any point P, it follows* that the potential 

 at P. due to the polarity of the dielectric, is 



-('-i)-('-.>. 



or 



M .A 'nfyiiftiU 



^(l^j\+ (}-l)vy that is 0, 



according as P is within S, within S' and without S, or without 

 S'. Hence the total potential will be, according to the position 

 ofP, 



„_(l-|)(U-U'), 





or V. -^<7./^i ■f>-my^[tf$nhD': 



Hence the sole effect of the dielectric C, on thfe state of A and 

 B, is to diminish the potential in the interior of the former by 

 the quantity 



(l_l)(U-U'). 



If the whole space between A and B be occupied by the solid 

 dielectric, the surfaces S and A will coincide, as also S' and B, 

 and therefore U=:V, U'=0. Hence the potential in the inte- 

 rior of A will be 



V 



■J. k ' f b^'r}-: 



or the fraction y of the potential, with the same charge on A, 

 and with a gaseous dielectric. From this it follows that, when 



* See Green's Essay, art. 12; or Math. Journal, vol. iii. p. 75. - 



