POLARIZATION BY REFLECTION. 1*J5 



perpcndicvilar to the ])lanc of reflection than in those coincident with that plane; 

 or, iu other words, that the reflected beam will be more or less polarized in the 

 plane of reflection. 



In order to estimate the amount of this polarization, we must take the diff'er- 

 once between the two terms in the value of R. And if we desire to find the 

 proportion of light polarized, we must divide this diflfcrence by the sum. Or 

 putting- P to represent this proposition : 



sin^(!+,'') V cos-(:— z?)' cos^f; — /)") — cos^Tj-I-^) _ , 



p — — ! — — - = ■ — ■ 1 32 1 



sin''^(.' — I)) I Q.Qi''\^i-Vi,V cos^i^t — />;+cos'^(^£+/>j "- '■' 



sin^(j+/>) ^ cos^(;— /))/ 



Reverting once more to the case of a wave polarized in azimuth a to the 

 plane of reflection, we shall perceive, by the formulte, that after reflection it is 

 Ktdl polarized, though not in the same plane as before ; for the rectangular 

 components of the molecular movement being unequally altered, their resultant 

 must have a new direction. In the expressions, 



sin':— /i) , tan(': — /)) . sin'c — p) cos'£ + />) . 



?;^-l_-^ — '■ . cos «, ?;'=_L ■ —. sma=_|- . -, ~\. -. ,. sma. 



-^sin;: + />) ^tan(i + /') ^^\\\\i-\-p) cos(^: — [>) 



the first is the molecular velocity normal to the plane of reflection, and the 

 second is the same velocity in the plane of reflection. The second divided by 

 the first is therefore the tangent of the inclination of the molecular movement te 

 the normal ; or of the resultant plane of polarization to the plane of reflection. 

 Putting this inclination = «', we have 



sina cos(t + /') ^ cos(c+/)) .^ , 



tana' = . ^ ;. =tana. . I o3.J 



cosa cos(f — (>) cos(f — p) 



If the reflected ray undergo reflection from a second mirror parallel to the 

 first, its incident azimuth will be a' ; and, after reflecuon, it will have another 

 azimuth a", of which the tangent will be 



,, ,C0S''£ + fl) COs'^^Tc+io) ■ rn* 1 



tana'=taua'. — . =tana, — ^ r. M*- 



cos(; — p) cos^(£ — p) 



And, as the law is manifest, we may say that after any number, n, of reflec- 

 tions, the tangent of the azimuth will be 



tana(«)=tanr//^-^^^i±^\" ' [35.] 



While cos(f+/') has a value — that is, while £+/> is more or less than 90^ — 

 tan("^a will also have a value ; or the plane of polarization of the wave will 

 not be brought, by any number of reflections, into absolute coincidence with the 

 plane of reflection ; but when £+/;=90'^, it will be so by the first reflection. 



When — '—. r is a small fraction, or at least not a large one, the plane of 



cos(; — p) 



polarization will, after a few reflections, be brought sensibly into the plane of 

 reflection. For instance, let i be 45°, and also a=45^. Then, for glass 



(index 1.50) /9 will be 28° nearly; and ^^^-— ^^ Avill be about jV I" this 



COS(£ — p) 



case one reflection will reduce the azimuth to IG*^ 42' ; two to 5° 9' ; three to 

 1° 32' ; four to 0° 28' ; and five to 0° 8^'. 



