Light at certain Metal-Liquid Surfaces. 235 



Theory. 



The theory of metallic reflexion in transparent liquids is 

 not developed in any of the ordinary works of reference, 

 though it is of course accessible in the original memoirs. 

 So it may not be out of place to give a brief resume of the 

 theory here. We start with the equation 



1 + tnn yfre 7A _ sin <f> si n % m# 



1 - tan f^ A ~ cos</> cos x' * ^ ^ 



where ^ is the azimuth of the restored polarization (the 

 angle whose tangent gives the ratio of the amplitudes ot the 

 two components of the reflected vibration when the incident 

 vibration is polarized in a piane making an angle of 4f>° 

 with the plane of incidence ; tan yjr is what we have c;illed the 

 " ellipticity " in the introduction), e is the Naperian base, 

 i= V — 1 } cj> and % are the angles of incidence and refraction 

 respectively ; while A is the phase difference between the 

 two components of the reflected vibration. 



In the case of the reflexion in a vacuum (or air) we have 

 the relation 



^-Wk, C2) 



sin x 



where K is the dielectric constant of the reflecting medium. 

 In the case of a metal this must be supposed complex and 

 the real part of >y/K is the index of refraction. If we 

 substitute for ^ in (1) from (2), replace the exponential by 

 its equivalent trigonometrical expression, and rationalize the 

 denominator of the left-hand side of (1), we obtain 



cos 2i/r( 1 4 i sin A tan 2yjr) _ sin cj> tan cf> 

 l-cosAsin2^ ' VK-sin^' 



This may be simplified by making the following substitu- 

 tions : 



sin A tan 2i|r= tan Q ; cos A sin 2^= cos P ; ) 



P f- . . (3) 



cos 2\jr~ cos Q sin P; S= tan - -sin <j>tau <fi ; 



which yield 



e^= =§ (4) 



* The derivation of this equation may be found in any standard text 

 on Optics ; a. g , those of Drude, Schuster, Wood. 



