from Crystallized Surfaces. 79 



an angle /3 determined by the formula 



Q-I y| 2 * 



tan j3 = cos (e 4 0) tan + 2 (# 2 - 2 ) sin sin ^ cos ^ - p - r . (2) 



When the ordinary ray disappears, the plane of polarization 

 of the reflected ray is inclined to the plane of incidence at an 

 angle )3 determined by the formula 



- tan j3' = cos (i + 0') cotan & 



,_ cos 20' . , sin 2 ?' 



sin cos 



. r r< . 



sin 9 sm( . -0 ) 



And when the angles /3, j3 r , become equal, the plane of polariza- 

 tion of the reflected ray becomes independent of the plane of po- 

 larization of the incident ray ; and the angle of incidence i, at 

 which this equality takes place, is the polarizing angle of the 

 crystal. Hence we have the equation of condition 



cos (i + 0) tan 9 + 2 (a 2 - 2 ) sin 9 sin i// cos 



sin (i <p 



orm 2 r/ m*n^ i 



+ cos (i -f d/) cotan 9' + (a 2 - b*) . -& sini// GOBI!/ -^r-. ; 



sin a sin(e 7 



to be fulfilled at the polarizing angle. 



Since i + 0, in this equation, is nearly equal to a right angle, 



put i + (j> = y + S, and 8 will be a small quantity. Draw PR 



an arc of a great circle perpendicular to ZOE, and let ZR = p, 

 PR - q. Then we shall find from equation (4), after various 

 substitutions and reductions, 



JTcos 2 q (cos 2 - cos ; where K= 



In deducing this value of S, the approximations were made 

 with a tacit reference to the case of reflexion in air from a com- 

 mon rhomb of Iceland spar. The coefficient K, in this case, is 

 equal to about nine degrees, and the resulting numerical values of 



