136 



STATIC ELECTRICITY 



Dealing in the same way with the tensions across DEF, they 



Q 2 



have a resultant outwards equal to g . 



When dr is very small, the resultant of these two is a pull inwards 



8K 



_1 \ _ (?dr 



(r+drf) 4Kr 3 ' 



This inward pull can only be neutralised by forces applied 

 round the rim of the shell. 



Let us assume at the rim a uniform pressure P upwards, i.e. 

 perpendicular to the lines of force. The total area of the rim is 

 2irrdr 9 so that for equilibrium 



and 



p _ 



_ Q 2 _ 



or a pressure normal to the lines of force equal to the tension 

 along the lines of force would maintain equilibrium. 



These equal values will maintain equilibrium in any 

 case. It is easy to show that when we have a field in which the le\el 



surfaces have double curvature tlu-i 

 equal values of the tension along the 

 lines and of the pressure tran-vi r^r to 

 the lines will suffice for equilibrium. 



Let ABCD, Fig. 91, be a small 

 rectangle on a level surface with its sides 

 in the planes of principal curvature. 



Let Oj O 2 be the centres of curva- 

 ture and let OjAsRj, O 2 B = R 2 . 



Let AO^ = <^ and BO 2 C = <f> r 

 The area of ABCD is l^R^^. 



The normals through "ABCD form 

 the tube of strain. Now draw a section 

 of the tube nearer to the centres of curva- 

 ture by <$, where S is very small, and let 

 A'B'C'D' be the comers of this section. 

 Its area is 



- 4 - if), neglecting -*_. 



H] IV ^I li 2 



If D X D 2 are the strains over the two surfaces ABCD, A'B'C'D 



respectively, 



or 



