440 Mr. G. H . Livens on the Electron 



3. The Optical Equations. 



The general optical equations for the metal are identical 

 with those contained in the former paper, and need not 

 therefore be developed in detail in the present instance. 



We may consider the general propagation of a plane 

 homogeneous wave train in the positive direction of the 

 axis of z, so that all functions specifying the propagation 

 are dependent on the time and space variables only through 

 the factor 



We may also assume that the waves are polarized so that 

 the electric force vector E is in the direction of the #-axis, 

 so that in the metal 



m 



The whole of the circumstances of the propagation are 

 then determined by the generalized quasi-index of refrac- 

 tion, which is in general a complex quantity (fi — ik) 9 the 

 real part (fi) of which determines the true refractive index 

 and the imaginary part (k) the absorption. From the 

 optical equations it is then easily deduced that 



ip 1 — aA 1— aA 9 



wherein 



__ 4r7T£ 2 Z, 



dmg + « ^o l + (/3<x) s 2 



e 2 /m 



and A=S- 9 



r n r — W n r—f 



which is a sum arising from the presence of the resonance 

 electrons bound by quasi-elastic forces in the interior of the 

 atoms, and is in fact taken over all of these electrons, each 

 with a proper frequency n r and resistance coefficient (mn' r ). 



The constant a which occurs in this relation is a numerical 

 constant of which an ideal estimate is 1/3. 



If we write 



and (yu r - ik r ) 2 = 1 + 1 __ aA , 



