( 386 ) 



Further 



2 U sin u sin itgi 

 dn m dn Wl-V,) = ^/jj^^MitJi- 



From this and from the value of tg {<pi — (pj^ follows 



. sin i tg i sin {(fi - <j)p) sin 2h /o i x i x 



Usinu-=. — r . • . • (^i) ) 



1 — cos {<pi — <pp) sm 2/i 



So from the restored azimuth h and the difference of phase 



<pi — (fj, at an arbitrary angle, Ucosu and Usinu or ncosa and k 



are to be derived for that angle. As ncos a^ [/n^ — sin^ i, we get 



afterwards n^ and k^ with the aid of (6) and (7). By means of them 



we can calculate tpi — (pp and h for every angle. 



9. As a rule we introduce the principal angle of incidence /, 

 for which (fi — </)^j = jr : 2. The restored azimuth at this angle is 

 called the principal azimuth H. As well from (20) and (21), as from 

 15) and (16) we may derive, when we add the index / to the 

 values of all the quantities for the principal angle of incidence : 



Uj =: sin Itg I , cos u^ ■=. cos 2 H . . . . (22) 

 According to (13) 



ki r= Ur sin rij = sin I tg I sin 211. (23) 



{n^ cos^ a) J =z nf — siV / = sin^ Itg^ I cos^ 2H 

 or 



7i/ = tg"- I{l—sin'Isin'2H) (24) 



We may also write (24) : 



nf-\-k/=:tfI^) (25) 



The optical constants 7io and kg are obtained from : 



n^' — ^/ " 71^^' — k/ = tg' I {1 — 2 sin-^ I sin^ 2 H) 

 or 



Wg h^ rzz [n cos u)i ki ■=. ^ sin^ Itg^ Isin 4: H . . . (26) 



10. When n^ and k^ have been given, we find by elimination 

 of ni and kj from the two first members of the two equations (26) 

 and (25) an equation of the sixth degree for the determination of 

 tg I. There may be given also approximating formulae for the deter- 

 mination of / and H from 7i„ and k^. From ii"^ — k^j=: n^^ — k^^ and 

 k \/if — sin^ 1= ?^o ^"o follows 



2w'^ = sin:'l-^n,'—k,'-]-[/{sin'I-7i,'—k,J-{-4:k,''sin'I 



Substituting this in ni--\- k/'=tg'I, we get: 



') This equation was ah'eady given by Ketteler, Wied. Ann,. 1, 241, 1877. 

 2) Ketteler calls this equation an analogon of tlie law of Brewster, VVied. 

 Ann 1, 242, 1877. 



