58 PROFESSOR KELLAND ON THE POLARIZATION OF LIGHT 



Now let us suppose that the plane of polarization of the reflected ray is in- 



p 

 clined by the angle a to that of incidence, we get tan a = ^-, (since R was 



measured upwards) 



/cosd). sin d>\ a T , ,, . 

 I - if- T-% ) cos 6 + -^ (-l) sin 

 \cos<i> sm<p,/ T 



.-. tana = -- ,. -r-r , - n -- 



(tan 9 cot <p t i) sin a 



, , , , . n T y (s 1) sin rf> sin <b. cos d> 

 = cos rf> + (>,) cot - =' v - y - ^ 



= - r , . a- 

 T sin (0 pj sin 6 



Now * varies as the mass of ether put in motion by the ray compared with 

 the same mass without the crystal. Also, it is such as to equal 1 when the ray 

 coincides with the wave ; and we can easily find the ratio of the masses in the 

 following manner. 



The mass outside the crystal has a common base at the surface of the crystal 

 with the mass inside. Also the slant heights corresponding with portions moved 

 during the same time, are in proportion to the velocities v, v of the wave with- 

 out and of the ray within the crystal. 



Lastly, the angles made by those slant heights with the common base are 

 the complements of <p and </> , which the wave and ray make respectively with 

 the normal. 



Hence we have the ratio of the volumes in motion within the crystal to that 



without = l ^r- 



v cos <p 



Let e be the angle which the ray makes with the wave: then (fig.) if T 

 be the place of the wave, T in the plane AT, will be that of the ray, and 

 XT =</> TT o =. 



But v -^ where v t is the velocity of the reave within the crystal, 



COS 6 



v. cos rf> sin rf> . cos rf> 



the ratio of the volumes moved = 



Now, by Spherical Trigonometry, it is evident that 



cos (f> = cos $>, cos e + sin fy, sin e cos 6, 



therefore the ratio of the volume in motion due to the ray within the crystal, to 

 that in motion due to the wave without, is 



e ~j! -- ( c s 0, c s e + sin (b, sin e cos 6). 



\ nf\c m ftf\c ^ ' ' 



sin (f> cos (/> cos e 

 Let A represent the ratio of the densities : then the ratio of the masses is 



rin^coB^ + Bin^cosetane (gee M < CuLLAGH) p. 29). 

 sin (p cos (f> 



But the value of s, when the ray coincides with the wave, is a multiple of 



