REFLEXION OF POLARIZED LIGHT. 253 



not what dependence we can establish between isolated 

 projectiles), the fact, and above all the laws, of inter- 



In the refracted ray the intensities of the residuary portions respec- 

 tively will be 



Here the second is always the greater: and the refracted ray con- 

 tains an excess of light polarized perpendicularly to the plane of inci- 

 dence. The difference or quantity of light polarized is the same as in 

 the reflected ray. Hence the light will be completely polarized at 

 any incidence for which either of the expressions (3.) or (5.) vanishes. 

 No value of i will make (5.) vanish, since we can never have i = r. 

 But the expression (3.) becomes = when i-\-r= 90. In this case 

 the light is completely polarized in the plane of incidence. But in this 



case we have also 



sin i . 



cos i sin r = - or tan i = i", 



which is Brewster's law ; also if i -\- r ]^>90 we have tan (i + rj. 



Also at this incidence the incident light is reflected, wholly polar- 

 ized in i; 5 is also transmitted wholly polarized in K. This is the cnse 

 referred to by Arago in the text. From (5.) also another remarkable 

 inference follows : if the reflexion be internal, or the ray be incident 

 on the second surface of a dense medium, we have r greater than i, 



or 



sin (i r) > 



that is, the phase of the reflected vibration is changed by 180 equiva- 

 lent to a difference of -J - in route, from what it would be in reflexion 



A 



at the first surface at the same incidence. This explains the supposed 

 assumption of the half undulation in Newton's rings. 



Again: if a polarized ray be incident on a reflecting surface with 

 its plane of vibrations inclined to the plane of incidence (i), at an 

 angle (a), its vibration (h) may be resolved into two, one in the plane 

 (i), and one perpendicular to it (K), in the ratio of sin a and cos a, 

 or after reflexion we shall have for the respective amplitudes (5.) 



and (3.) 



k' sin a, and hi cos a. 



These by composition will give a resultant ray polarized in a plane 

 (p), inclined to (i) by angle (/3), and we have from the formulas (5.) 



and (3.) 



cos (i + r) 

 tan 8 = tan a - -~- 



cos (i r) 



