LAW OF MALUS. 129 



In tliis case p is the angle of incidence and : tlie angle of refraction for the 



second surface, the index of refraction being - And we have — 



n. 



sine . . . _ 



tan/? sm£ = sinc = sm/); or, sm.' = cos/), and £ + /5 = 90'^ 



COSf» • 



Wc have seen that when the two polarized rays into wliich a sinjjle ray of 

 common light is divided by double refraction in passing through a rhomb of 

 Iceland spar fall upon a second similar rhomb, they are both of them subdi- 

 vided in most of the positions of the second rhomb ; but that the intensities of 

 the rays of each pair are unequal, except when the prmcipal planes of the 

 rhombs differ in azimuth 45°, and that one membcii.- of each pair disappears en- 

 tirely when the principal planes arc coincident or normal to each other. The 

 inequality of intensity is variable, and is dependent on the angle between the 

 principal planes. If one ray of either pair be observed through all its varia- 

 tions, it will be found to begin from zero of intensity, to increase regularly in 

 brightness for 90°, and then to diminish through the second 90°, to zero again. 

 The other member of the same pair passes through a similar series of changes, 

 but its maxima correspond in azimuth to the minima of the first, and its minima 

 to the maxima of the first. 



A ray which has been polarized by reflection possesses the same character 

 as those which have been produced by double refraction in Iceland spar; and 

 accordingly, if such a ray be transmitted through a doubly refracting rhomb 

 which is turned in azimuth in the manner just described, it will be divided into 

 two rays which will alternately increase and diminish in intensity ; and of which 

 one will become zero in the azimuth 0° or 90° between its plane of polarization 

 and the principal section of the rhomb. Assuming the united ijitensities of the 

 two rays into Avhieh a single one is thus divided by double refraction to be equal 

 to the total intensity of the original ray, JMalus inferred that their several inten- 

 sities should vary as the squares of the sines and the cosines of the azimuth. 

 Thus, if /be put for the total original intensity and a for the azimuth, reckoned 

 from the position- of coincidence of the plane of polarization with the principal 

 section of the rhomb, then the ordinary ray would have the intensity Jcos'a; 

 and the extraordinary, Jsin^a. These values fulfil the condition of constancy 

 of sum ; since — 



I cos^a + I sin^a = I. 



If a ray which has been polarized by reflection fall, at the polarizing angle, 

 upon a second mirror of transparent glass with parallel faces, it will be divided 

 into two rays ; one of which will be reflected and the other transmitted. When 

 the second mirror is turned in azimuth around the incident ray, these two de- 

 rivative rays will undergo changes of intensity somewhat resembling those 

 which have just been described as produced by double refraction. When the 

 two planes of reflection are coincident, the intensity of the reflected ray will be 

 maximum, and that of the transmitted ray, minimum. This minimum will not, 

 however, be zero. When the two planes differ in azimuth 90°, the intensity of the 

 transmitted ray Avill be maximum, and that of the reflected ray, minimum. 

 This minimum will be zero ; and the simultaneous maximum of the transmitted 

 ray will be equal to the total intensity of the incident light. The alternations 

 in this case resemble, therefore, to a certain extent, those previously described 

 as produced by double retraction ; but they are not represented by the law of 

 Mains. 



The plane of j)olarization — an expression which we have just used without 

 defining it — is the plane in which a polarized ray is capable of being reflected 

 at the polarizing angle. Accordingly, when a ray of common light is polarized 

 by reflection, the plane of incidence and reflection is itself the plane of polariza- 

 tion. 



In the arrangement of two mirrors, as above described, when the second 

 9s 



