108 Mr. G. H. Livens on the Number of 



The quantity A which appears in this formula arises 

 •entirely from the resonance electrons, which, for other reasons, 

 are supposed to exist in the atoms, and which will therefore 

 be effective in modifying the simple direct action of the 

 electric field in the incident radiation. It consists, as in the 

 usual theory of dispersion, in dielectrics mainly of terms of 

 type 



a _ v e V m 



r n/ — re —imir 



the sum X being taken per unit volume over all these 

 resonance electrons. 



The constant a represents an absolute constant of which 

 an ideal estimate gives a = 1/3. In any real case, however, 

 and particularly in solid media, this number may be widely 

 departed from ; to this extent alone the theory provides a 

 -distinctly uncertain result, although it must be said that a 

 sufficient knowledge of the optical properties of the metal 

 over a wide range in the spectrum would enable us to obtain 

 a precise estimate even of this constant. 



We shall now make the assumption that there is no appre- 

 ciable absorption due to the resonance electrons. This as- 

 sumption is fully justified if we are using light-waves of 

 frequency different from those of the free oscillations of the 

 resonance electrons. We may then take A to be real, and 

 so we put 



and fip is the index of refraction of the metal with all the 

 freely moving electrons removed. We have, then, that 



2 



= l + a(^ 2 -l) 



1-aA 



We now separate C into real and imaginary parts, and write 

 C = d + iC 2 , 



so 



that 



3m V 7r 3 .. rt , 



hilst 



n _ 87rp~Ne 2 ql r 



rrl 2} 



~o i + £ip- 



V?J 1 + ^ 2 ' 



