144 The State of the Medium in the Electrostatic Field [OH. vi 



THE STRESSES IN AN ELECTROSTATIC FIELD. 



161. If an infinitesimal charged particle is introduced into the electric 

 field at any point, the phenomena exhibited by it must, on the present view 

 of electric action, depend solely on the state of stress at the point. The 

 phenomena must therefore be deducible from a knowledge of the stress- 

 quadric at the point. The only phenomenon observed is a mechanical force 

 tending to drag the particle in a certain direction namely, in the direction 

 of the line of force through the point. Thus from inspection of the stress- 

 quadric, it must be possible to single out this one direction. We conclude 

 that the stress-quadric must be a surface of revolution, having this direction 

 for its axis. The equation of the stress-quadric at any point, referred to 

 its principal axes, must accordingly be 



^p+^<y-4-r 2 ) = i ........................... (83), 



where the axis of f coincides with the line of force through the point. Thus 

 the system of stresses must consist of a tension 7? along the lines of force, 

 and a tension P z perpendicular to the lines of force and if either of the 

 quantities J? or % is found to be negative, the tension must be interpreted 

 as a pressure. 



Since the electrical phenomena at any point depend only on the stress- 

 quadric, it follows that R must be deducible from a knowledge of 7? and 7?. 

 Moreover, the only phenomena known are those which depend on the 

 magnitude of R, so that it is reasonable to suppose that the only quantity 

 which can be deduced from a knowledge of 7? and 7? is the quantity R 

 in other words, that 7? and ^ are functions of R only. We shall for the 

 present assume this as a provisional hypothesis, to be rejected if it is found 

 to be incapable of explaining the facts. 



162. The expression of /? as a function of R can be obtained at once 

 by considering the forces acting on a charged conductor. Any element dS 



R 2 

 of surface experiences a force g- dS urging it normally away from the con- 



ductor. On the present view of the origin of the forces in the electric field, 

 we must interpret this force as the resultant of the ether-stresses on its two 

 sides. Thus, resolving normally to the conductor, we must have 



where (7?) E , (/J) denote the values of 7? when the intensity is R and 

 respectively. Inside the conductor there is no intensity, so that the 

 stress-quadrics become spheres, for there is nothing to differentiate one 

 direction from another. Any value which (7?) may have accordingly arises 



