1884.] equations of the electromagnetic field. 121 



movement of electricity, the phenomenon which we have called 

 electric displacement is a movement of electricity in the same 

 sense as the transference of a quantity of electricity through a 

 wire is a movement of electricity." 



Helmholtz considers the electric polarization set up in the 

 medium by the electric force. 



Each element of the medium becomes polarized, opposite elec- 

 tricities shewing themselves at the opposite ends. 



Let us suppose that in the element of volume dv there are 

 two quantities E and — E of electricity at a distance s apart, 

 these two quantities having each been moved the distance \s by 

 the action of the force. Then, according to Helmholtz, we may 

 call Es the electric moment of the element, and the ratio of the 

 electric moment to the volume of the element is the intensity of 

 the electric polarization in the element in the direction of the 

 resultant force. This intensity of the polarization can be resolved, 

 and if £, rj, £ be its components then Helmholtz puts £ = eP. 

 P as before being the electromotive force and e the dielectric 

 constant of the medium. 



To compare the two theories it becomes necessary then to 

 determine the relation between f and £. The quantity £ like f 

 is a surface density, being the surface density of the electricity 

 induced by the action of the electromotive force on the face of 

 the element normal to Ox, while / as we have seen is the surface 

 density of a distribution which will produce over the same surface 

 the actual force. 



Now according to Maxwell, if V be the potential at any point 

 in the dielectric and p the density of the free electricity, then 



Ky 2 V+47rp = 0, 



\dx) \dyj \dz 



According to Helmholtz, 



(1 + 47re) v 2 V + 47rp = 0. 



The potential due to a quantity of electricity E placed at a 

 point in the medium at distance r is on the two theories, 



E/Kr and Ef{\ + 47re) r. 



Thus we see that K= 1 4- 47re, 



, - K „ 1 + 47T6 „ 

 aucl J = ~a Z — —A £ 



We may compare these equations with those in Poisson's theory of 

 induced magnetization (Maxwell, Vol. II. § 426, etc.). 



