POLARIZATION BY REFLECTION. 193 



If we suppose the iucidence perpendicular, we shall have, as Lefore, 

 t''^= ( "TTi ) ' ^^^ "'^~ 1 "XT ) • -^^ incidence 90 \ we Lave again, 

 ^,^ = (lJLZl\ = 1 ; and u''=0. 



Tlie agreement of these formulas with those obtained in the other case, is 

 what ought to be expected, since, at a perpendicular iucidence, the direction of 

 molecular movement can have no influence on reflection; and at 90° impact ceases. 

 But if we examine the first value of v' given above, we shall perceive that it 

 does not constantly increase with the increase of the incidence ; for the denom- 

 inator, tan (£+jo). becomes infinite when ;-f/>=90^; and at this incidence v'=0, 

 or there is no reflection. If then, in the originally incident beam, there had 

 been a succession of waves, some of them polarized in the plane of incidence, 

 and the rest polarized at right angles to that plane, all this latter class of waves 

 would, at this particular incidence, be transmitted, while a portion of the others 

 would be reflected. The incident light, from the mixture of the two classes of 

 waves, would be imperfectly polarized, or not polarized at all : but the reflected 

 light would be wholly polarized in the plane of reflection. 



In the result just reached, we see a reproduction of the law experimentally 

 established by Brewster, viz., that, at the polarizing angle, the transmitted ray 

 is at right angles to the incident ray, or c-{-p=:90\ 



If we now take the case of a wave whose plane of polarization is in any azi- 

 muth to the plane of reflection between 0° and 90 ''j we may apply the principles 

 already illustrated, by decomposing its molecular movements into comnonents, 

 one of which shall coincide with the plane of reflection, and the other with the 

 reflecting surface. If the given azimuth be a, the azimuth of the molecular 

 movements will be 90° — a. The molecular movement in 'Oao, i)lane of rcjlectioa 

 will therefore be cos(90° — a)^=sina ; and that in the reflecting surface will be 



That is to say, at this incidence the entire molecular uiorement normal to the plane of reflec- 

 tion is transmitted. 



In this case the condition \-\-v'=u' is always fulfilled. Vv^hen vibration is in the plane of 

 reflection it is fulfilled only for the peii^endicular incidence. At other incidences the fourth 

 principle of McCullagh and Neuman, quoted above, necessarily involves the truth of Fiesnel's 

 assumption for this case, viz : 



(l+r')cos<=w'cosp. 



When n is less than 1, these formula3 fail for incidences beyond the limitiuf au"-le of total 

 reflection. 

 The formula in the text admit of reductions similar to the foregoing. They thus become 



1. For vibrations in the plane of reflection — 



sin2f — sin2p 4sinpcosf 



sin2t-f-sin2p' sin2i-f-sinyp* 



2. For vibrations normal to the plane of reflection — 



, sin2(i — p\ ^_4siupcostcos(i — p) 



sin2i-|-sin2p' sin2i+siu2/3 



And for convenience of discussion : 



1. For vibrations in the plane of reflection — 



«cost-^cosp ^COSi 



MCOSt-j-COSp ' »iCOSi-{-COSp * 



2. For vibrations normal to the plane of reflection — 



, ncosp — cost ^ 2cost 



HCOsp-|-cost " mcosp-f-cost 



The first of these values of v becomes zero at tlie polarizing angle, and is positive for all 

 higher incidences. 



13 s 



