REFLEXION AND REFRACTION OF POLARIZED LIGHT. 177 



(189) The reflected light will be completely polarized, 

 when one of the portions of which it consists vanishes ; for, 

 in this case, the whole of the reflected light will be polarized 

 in a single plane. It is easy to see that the first portion, 

 which is polarized in the plane of incidence, can never vanish. 

 The second part vanishes, when 



i + r = 90, 



the denominator of the fraction becoming infinite ; the re- 

 flected light then contains only the other portion, and is 

 therefore completely polarized in the plane of incidence. From 

 the foregoing relation it follows that 



sin i i i 



cos ^ = sm r = , and tan i =ju. 



/* 



i. e. the tangent of the angle of polarization is equal to the re- 

 fractive index. Thus the beautiful law, which Brewster had 

 inferred from observation, is deduced as an easy consequence 

 of Fresnel's theory. 



When i + r is greater than 90, i. e. when the angle 

 of incidence exceeds the polarizing angle, the expression for 

 the amplitude of the reflected vibration w, changes sign, the 

 light being polarized perpendicularly to the plane of inci- 

 dence. This change of sign is equivalent to an alteration 

 of the phase of the reflected vibration by 180, as the in- 

 cidence passes the polarizing angle ; and the circumstance 

 explains the remarkable fact noticed by Arago, namely, that 

 when Newton's rings are formed between a lens of glass v 

 and a metallic reflector (the incident light being polarized 

 perpendicularly to the plane of reflexion), the rings change 

 their character as the incidence passes the polarizing angle 

 of the glass, the black centre being transformed into a white 

 one, and the whole system of colours becoming complementary 

 to what it was before. Sir Gr. B. Airy was led to anticipate this 



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