from Crystallized Surfaces. 79 



an angle /3 determined by the formula 



sin" i 



tan 8 = cos (i 4 rf>) tan 9 + 2 (a 2 - S 2 ) sin 9 sin \L cos i// - r . r . (2) 



sin (i - 0) 



When the ordinary ray disappears, the plane of polarization 

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

 angle j3 determined by the formula 



- tan )3' = cos (i + $') cotan 0' 



. . cos20 / . , , sin 2 / 



+ (rt 2 - 2 ) . 7J7- sm ^/ cos u/ - - 7-. (3) 



sin u sin (i . $ ) 



And when the angles j3, /3', 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 /, at 

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

 crystal. Hence we have the equation of condition 



cos (/ + 0) tan 9 + 2 (a z - b 2 ) sin 9 sin $ cos 



,. .. fl , . 2 72N . , , 



+ cos (t+9) cotan + ( 2 - 1> 2 ) : ^ sm\L cos\L -^ 



' 



sn -fj) 

 sin 2 / 



to be fulfilled at the polarizing angle. 



Since i + <}>, in this equation, is nearly equal to a right angle, 



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







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

 PJS = q. Then we shall find from equation (4), after various 

 substitutions and reductions, 



(ffi _l$\ ( ] , 7.2N 



(7 (cos 2 - cos 2 j>) ; where ^= { ( ^ , >. (5) 



In deducing this value of 8, 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 XT, in this case, is 

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



