( 4Öf) ) 



Physics. — ''Derivation of the fundamental equation.'^ of metallic 

 reflection from Cauchy's t/ieorif'. By Prof. R. Sissingh. (Com- 

 municated by Prof. H. A. Lorentz). 



1. It has been pointed out in a previous paper ^) tliat the tlieories 

 of metallic reflection drawn up by Cauchy, Kettklkr and Vonrr and that 

 by Lorentz lead to identical results. It must therefore also be possible in 

 the theory of Cauchy to derive the two relations which the three last 

 theories furnish between index of refraction and coefticient of absorp- 

 tion for normal and oblique iiu-idence of the light that penetrates 

 into a metal, the so-called fundamental equations. These fundamental 

 equations may lirst be obtained by paying regard to the connection 

 of the quantities which the Iheory of Cauchy and the other theories 

 introduce for the description of the phenomenon. Cauchy determines 

 the so-called comple.v angle of refraction r by sin r = sin i : öe'~ and 



cos r =z oe''"''). From this follows 1 — -r ^ oV'"', so that: 



Ö"-' cos 2x— c/ a' cos 2 (t + CO) + sin'' I . . . . (1) 

 Ö* sin 2 T = o' ö' si7i 2 {r -{- et)) (2) 



If Ave pay regard to tlie relations between ^t, t and y/„ and /i„, 

 index of refraction and coefficient of absorption for normal incidence, 

 and to the equations (17) and (18) of the preceding papei- "), the 

 ecpiations (1) and (2) appear lo be nothing but the fundamental 

 ecjuatious, given in e(| nation (G) and {7) of the previous paper. 



2. On account of the close connection between the theories of 

 metallic reflection it must, however, be also possible, to derive these 

 fundamental equatio::s from Cauchy's theory without paying attention 

 to the coimection with the others. The fundamental idea ofCAUCiiv's 

 theory is the introduction of a complex index of refraction. Denote 

 this again l)v n^ -\- il-g = ae'-', so that 



n^ -=10 cost , k^^z=:asiiir (3) 



and 



sin r = sin i : Oe^~ (4) 



while we |iut 



cos r = Qe*^'" (5) 



Let the A'i^-plane of a rectangular system of coordinates be the 

 plane of incidence of the light })enetrating into the metal, and the 

 J"yf-plane the bounding plane of the metal, the A'-axis being directed 



1) Sissingh, These Proc. VIII p. 377. 



2) Loc. cit. p. 385. 



