34 



STATIC ELECTRICITY 



will hold good for every elementary cone drawn from () to cut S- 

 and therefore an element of charge g without S on the whole makes 

 no contribution to /NdS, and only the charge within S counts. 



We have taken the simple case in which each cone drawn from 

 a point inside cuts the surface once, and each cone from the out- 



Fio. 25. 



side cuts it twice, but it is easy to sec that the ioult> arc true if 

 the surface is re-entrant, so that it is cut any odd number of 

 times in the first case and any even number of times in tl it- 

 case. 



We shall now apply Gauss's theorem to tubes of force. 



If a tube of force starts from a given charge it either 

 continues indefinitely, or if it ends it does so on an equal 

 and opposite charge. Let a tube start from a (juantity 7 on 

 an element of surface S r Fig. 26'. In dcM-ribing the t ; art ing 



from q we imply that within S A the intensity is /no. 



Suppose that the tube continues some distance from S A and 



FIG. 26. 



that it ends at P, as represented in the figure, without reaching 

 any other charge. Prolong the tube within S r make another 

 closed end, and to the closed surface thus formed applv Gauart 

 theorem. Over the whole of this surface N = 0, for alon^ the 



