( 385 ) 



system of two deterniijiing quantities is chosen, however, the ampli- 

 tudes and phases of the two components of the reflected light pola- 

 rized in and normal to the plane of incidence, whether calculated 

 in one way or in the other, will have the same values. The two 

 systems of formulae are therefore identical. 



Cauchy ^) calls the so-called complex index of refraction de'"^ . This 

 quantity being represented here by n^ -\~ ih^, 6 cos r = n^, asi?i r = k^. 

 The auxiliary quantities q and a> have been introduced by Cauchy 

 for the determination of the so-called imaginary angle of refraction r, 

 determined by siii r = sin i : {n^ -\- ik^) '). In order to express 9 and 

 CO in the quantities used above, it may be observed that: 



cos^ r sin^ i : sm* r = cos^ r {71^ -{- ikj' = (n^ -f ^^o)" — ^^^^ ^ 

 or with the aid of (6) and (7): 



cot r sin i ^=in cos a -\- ik. 



As Cauchy gives : 



cos r = ^e''" ^) and n^ -j- ^K = ^^"' » n cos a -}- ik is equal to ^TeC'^H-") 



or 



n cos a = U cos u := Q<J cos {r ~\- 0} ) . . . . (17) 



k^ U sin u =. QO sin (t-{-0)) (18) 



The equations (17) and (18) allow us to deduce our auxiliary 



quantities from those of Cauchy and reversely '). 



8. According to § 7 : 



^ , 2 Usin itg i , 2U sin i tq i 

 cos 2/i = cos u -— , tq ((pi — (pr,) =. sm u — . 



U'-^sin' i tgH ' "^ ^^ ' ^^' U'—sinH tgH 



These two equations may serve to determine U and u, and from 

 this the optical constants ??,„ and k^ with the aid of (13), (6) and (7). 

 From the values of cos2h and t</ ((fi — (p^) follows: 



U^ — sin^ i tg^ i 



sin 2/i cos ((pi — tf)„) 



v^' ^}'^ U^ ^sin'itg'i 



or 



2 sin^ i tg^ i 

 l-nn2hcos(,f,-,f„)=-^^^^-^—. . . . (19) 



From (19) and the value of cos 2h follows : 



sin i tq i cos 2 A 



Ucosu = - (20) 



1 — cos {<fil—<Pp) sin 2/i 



V LoRENTz, I.e. Theorie der terugkaatsing en breking, p. 164, Schlömilch's Zeitschr., 

 23, pg. 206. EisENLOHR, p. 369. 



2) See note 1 on the preceding page. 



") Cf. also Ketteler, Wied. Ann. 1, 24-2, 1877; 33, 212, 1884. Formulae for 

 the calculation of p and a' are given by Lorentz, Theorie der Terugkaatsing on 

 Breking, p. 164, 165, Eisenlohr, Pogg. Ann., 104, 370, 1858. 



26 



Proceedings Royal Acad. Amsterdam. Vol. Vlll. 



