of Light by thin Layers of Metal. 453 



derivation of consequences practicable and suitable for compa- 

 rison with experiment. 



The three media to be considered shall, according to the order 

 of succession, be designated as the 1st, 2nd, and 3rd; and the 

 quantities referring to them shall have the corresponding indices 

 3, 2, 3. Accordingly let \ and X 3 be the wave-lengths, « x and 

 « a the angle of incidence in the first and the angle of refraction 

 m the third medium. The sines and cosines of these angles will, 

 for the sake of brevity, be denoted by s v c v s 3 , c 3 . The wave- 

 length \ 2 and the angle of refraction a 2 , for the second, metallic 

 medium, are, it is well known, according to the notation intro- 

 duced by Cauchy, imaginary. If, then, we put 



X 2 =/ 2 e-« 

 and denote the complex values of sin a 2 and cos « 2 by 



in which / 2 , s 2 , c 2 are real positive quantities, and e, u real angles 

 lying between and ^, it is well known we shall We 



Ji _ £? = % 



and the complex refraction-ratios for the passage out of the first 

 or third into the second medium will be expressed by 0^ or 3 e ei , 

 where 



*,= !« = £.', e 3= s J> = h. 



S 2 ^2 $2 ^2 



The quantities s 2 , c 2 , e, u are therein connected by the relations 

 ^cos2e + ^cos2w = l, 

 s\ sin 2e — c\ sin 2u = 0, 



which are always to be identically fulfilled. 

 For abbreviation, further, for v=l or 3, let 



U , = 1 +^ cos ( e + w )+^ ^ = 1 + 2^008 (e-tt) + q;. 

 V v = 1 - 2 ^ c °s(e + w) + ^ % v =l-2q v cos(e-u) + (g. 

 Denoting the thickness of the lamellae by A, and putting 



, T 27TC 9 COS (e + u) ^ 2nc. 2 sm(e+u) 



L = — J 1 ~- A > D=<5 5 .A, 



there result, for the amplitudes a' and a m and for the retarda- 

 tions of phase d' and d m of the ray reflected in the first medium 

 and of that refracted m the third medium, the following ex- 

 pressions : — G 



I. For light polarized parallel to the plane of incidence : 



