428 The Theory of Aether and Electrons in the 

 which vanishes by virtue of the principle of conservation of 



div 8 = 0, (V) 



electricity. Thus 



or the total current is a circuital vector. Equations (I) to (Y) 

 are the fundamental equations of Lorentz' theory of electrons. 



We have now to consider the relation by which the polari- 

 zation P of dielectrics is determined. If the dielectric is 

 moving with velocity w, the ponderomotive force on unit 

 electric charge moving with it is (as in all theories)* 



E' = E + [w . B ]. (1) 



In order to connect P with E', it is necessary to consider the 

 motion of the corpuscles. Let e denote the charge and m the 

 mass of a corpuscle, (, *?, ) its displacement from its position of 

 equilibrium, k* (, 77, ) the restitutive force which retains it in 

 the vicinity of this point ; then the equations of motion of the 

 corpuscle are 



ra + A- 2 = eE x ' t 



and similar equations in 17 and . When the corpuscle is set in 

 motion by light of frequency n passing through the medium, 

 the displacements and forces will be periodic functions of nt 

 say, 



Substituting these values in the equations of motion, we obtain 

 A(Jc* - mn z ) -= eE, and therefore ? (k z - tun*) = eE' x . 



Thus, if N denote the number of polarizable molecules per unit 

 volume, the polarization is determined by the equation 



* = Ne (g, TJ, ?) = JVVE7(& 2 - m?i 2 ). 



In the particular case in which the dielectric is at rest, this 

 equatio^ gives 



= (l/47rc 2 )E + P = (l/47rc 2 )E + Ne*E/(k 2 - mw\ 

 But, as we have seen,f D bears to E the ratio ^u 2 /47rc 2 , where ^ 



*Cf. p. 365. tCf. p. 281. 



