Freedom of Electrons in Metals. 313 



proportions determined by their olmiic conductivities alone. 

 For that type of radiation, the square of: the qua&i-index. of 

 refraction is therefore a pure imaginary quantity. 



The opposite extreme case, that of radiation of period rapid 

 compared with the times of undisturbed motion of the elec- 

 trons, is also of interest. It will appear that, if the effective 

 electrons are free, the square of the quasi-index, must be a 

 real negative quantity. The optical determinations of Drude 

 indicate that this is not very far from being true with light- 

 waves for some of the nobler metals; in the case of white 

 metals such as silver, the property, moreover, persists over a 

 considerable range of period, though at length it fails. Thus, 

 both on this ground and by reason of the shorter period, 

 over a rather wide range of optical period the electrons are 

 perhaps not far removed from being free. Moreover, theories 

 of ordinary complete metallic conduction have been developed 

 with some promise which ascribe it to electrons entirely free. 



When the electrons in the optical problem are supposed to 

 be virtually free, then for each of them, of inertia m and 

 charge e, under electric force (P, Q, R), 



rn,(.v,y,z)=eQ?, Q, R) ; 



thus if there are N ; of them per unit volume, and (u r , r\ ic r ) 

 is the current of conduction, 



|(»',,,>')=N'fL(P,Q,R), 



the sign of the charge e not entering if m is the same for all. 

 Consider a plane-polarized wave-train travelling along z, 

 with electric vector (P) along x and magnetic vector (ft) 

 along- y. Its equations of propagation are, by the circuital 

 relations of Ampere and Faraday, 



where . . , , _ d~P 



at 



the last term representing the quasi-current of a?ther strain 

 which will prove to be here negligible. Thus 



d 2 P , du 

 ^ = ^dt 



e 2 d 2 P 



m at' 



which determines the mode of propagation. 



Phil. Mag. S. 6. Vol. 14. So. 80. Aug. 1907. Y 



