161-164] Stresses in Electrostatic Field 145 



simply from a hydrostatic pressure or tension throughout the medium, and 

 this cannot influence the forces on conductors. Leaving any such hydrostatic 

 pressure out of account, we may take (/J) = 0, and so obtain (7?)^ in the 

 form 



163. We can most easily arrive at the function of R which must be 

 taken to express the value of P 2 , by considering a special case. 



Consider a spherical condenser formed of spheres of radii a, b. If this 

 condenser is cut into two equal halves by a plane through its centre, the 

 two halves will repel one another. This action must now be ascribed to the 

 stresses in the medium across the plane of section. Since the lines of force 

 are radial these stresses are perpendicular to the lines of force, and we see 

 at once that the stress perpendicular to the lines of force is a pressure. To 

 calculate the function of R which expresses this pressure, we may suppose 

 b a equal to some very small quantity c, so that R may be regarded as 

 constant along the length of a line of force. The area over which this 

 pressure acts is 7r(b 2 a 2 ), and since the pressure per unit area in the 

 medium perpendicular to a line of force is J^, the total repulsion 

 between the two halves of the condenser will be I^TT (b 2 a 2 ). 



The whole force on either half of the condenser is however a force 27r<r 2 

 per unit area over each hemisphere, normal to its surface. The resultant of 

 all the forces acting on the inner hemisphere is ?ra 2 x 27r<r 2 , or putting 

 27ra 2 cr = E, so that E is the charge on either hemisphere, this force is E*/2a?. 

 Similarly, the force on the hemisphere of radius b is E*/2b*. Thus the re- 



sultant repulsion on the complete half of the condenser is \E* ( j- ) . Since 



\ / 



this has been seen to be also equal to I^TT (6 2 a 2 ), we have 



on taking a = b in the limit. 



Thus in order that the observed actions may be accounted for, it is 

 necessary that we have 



Moreover, if these stresses exist, they will account for all the observed 

 mechanical action on conductors, for the stresses result in a mechanical force 

 27r<7 2 per unit area on the surface of every conductor. 



164. It remains to examine whether these stresses are such as can be 

 transmitted by an ether at rest. 



j. 10 



