56 PROFESSOR KELLAND ON THE POLARIZATION OF LIGHT 



Multiplying this equation by the former, we have 



(cos & sin & cot cos (f) 0') (cos 6 cos + <p' cot a sin 0) = 



(cos 6 cot u cos (f> 0' + sin 6) (sin & cos (p + (f)' cot a + cos &) 



or cos 6 cos & cos + 0' cot a + sin Q sin & cos<f> 0' cot w= 



cos 0cos # cos0 (f)' cot u sin 0sin #cos0 + 0'cot a, . . . (12) 



or (cos 6 cos & + sin sin &) (cos + 0' + cos 0' tan a cot ). 



7T 



The first factor gives 6=& + -~ which shews that the two planes of vibration coin- 

 cide when ta is given. This result is interpreted by saying, that if the incident 

 light is polarized, the refracted light is polarized also. The other factor will be 

 zero when the light incident is common light, that is when *> is indeterminate ; 

 and, further, it is evident that both its terms will be separately zero or tana=0, 

 and cos (0 + 0')=0. Of these the former shews that the plane of polarization coin- 

 cides with that of incidence ; and the latter, that the value of the polarizing angle 

 is determined from the circumstance that the angles of incidence and refraction 

 are complementary to one another, which is the well-known law obtained by expe- 

 riment. 



If we suppose, as FRESNEL does, that the transmitted light consists of vibra- 

 tions polarized in two planes at right angles to each other, we have 6 = #=0, 



sin0 + 0' cos0- 0' _ _, sin 0-0' cos + 0' 



\. J. . -r~, "T . XV ^Z X ; -7 -7 , 



sin cos (p sin cos <p 



9T ,_ T , sin0 + 0' OT? ,_ T , sin (ft -ft' 



i I. 1 i -, . ,- * 4 XV Jl : -77 -r , 



sin <p cos <p sin (p cos <p 



sin (p (j>' cos ft + < 

 ' cos (> ( 



= _ j tan 0-0' ^ (FRESNEL'S result.) 



tan + 0' 



2 sin 0' cos 

 sin + 0' cos 0" 



,-. 

 Do.) 



T'= 



sin 



Thus all FRESNEL'S four results are contained in our equations as particular 

 cases. 



