128-131] Gauss' Theorem 119 



the space enclosed by the surface, leaves it again, will add no contribution to 

 II PcosedS, its strength being counted negatively where it enters the 



surface, and positively where it emerges. A tube which starts from or ends 

 on a charge e inside the surface S will, however, supply a contribution to 



II P cosedS on crossing the surface. If e is positive, the strength of the 



tube is e ; and, as it crosses from inside to outside, it is counted positively, 

 and the contribution to the integral is e. Again, if e is negative, the strength 

 of the tube is e, and this is counted negatively, so that the contribution is 

 again e. 



Thus on summing for all tubes, 



where E is the total charge inside the surface. The left-hand member is 

 simply the algebraical sum of the strengths of the tubes which begin or end 

 inside the surface ; the right-hand member is the algebraical sum of the 

 charges on which these tubes begin or end. Putting 



PR 



4-7T 



the equation becomes 1 1 KR cos edS = 4<7rE. 



The quantity R cos e is, however, the component of intensity along the 

 outward normal, the quantity which has been previously denoted by N, so 

 that we arrive at the equation 



(61). 



When the dielectric was air, Gauss' theorem was obtained in the form 



NdS = 4>7rE. 



is 



Equation (61) is therefore the generalised form of Gauss' Theorem which 

 must be used when the inductive capacity is different from unity. Since 



dV 

 N= -^ , the equation may be written in the form 



~dn 



131. The form of this equation shews at once that a great many results 

 which have been shewn to be true for air are true also for dielectrics other 

 than air. 



It is obvious, for instance, that V cannot be a maximum or a minimum 

 at a point in a dielectric which is not occupied by an electric charge : as 



