120 Dielectrics and Inductive Capacity [CH. v 



a consequence all lines of force must begin and end on charged bodies, 

 a result which was tacitly assumed in denning the strength of a tube of 

 force. 



A number of theorems were obtained in the discussion of the electrostatic 

 field in air, by taking a Gauss' Surface, partly in air and partly in a con- 

 ductor. Gauss' Theorem was used in the form 



but we now see that if the inductive capacity of the conductor were not 

 equal to unity, this equation ought to be replaced by equation (61). It is, 

 however, clear that the difference cannot affect the final result ; N is zero 

 inside a conductor, so that it does not matter whether N is multiplied by K 

 or not. 



Thus results obtained for systems of conductors in air upon the assumption 

 that Coulomb's law of force holds throughout the field are seen to be true 

 whether the inductive capacity inside the conductors is equal to unity or not. 



The Equations of Poisson and Laplace. 



132. In 49, we applied Gauss' theorem to a surface which was formed 

 by a small rectangular parallelepiped, of edges dx, dy, dz, parallel to the 

 axes of coordinates. If we apply the theorem expressed by equation (61) to 

 the same element of volume, we obtain 



......... (62), 



where p is the volume density of electrification. This, then, is the genera- 

 lised form of Poisson's equation : the generalised form of Laplace's equation 

 is obtained at once on putting p = 0. 



In terms of the components of polarisation, equation (62) may be written 



while if the dielectric is uncharged, 



+ = 0... ...(64). 



dx d dz 



Electric Charges in an infinite homogeneous Dielectric. 



133. Consider a charge e placed by itself in an infinite dielectric. If 

 the dielectric is homogeneous, it follows from considerations of symmetry 

 that the lines of force must be radial, as they would be in air. By application 



