Reflexion and Refraction. 119 



2. When X = 0, the axis lies in the face of the crystal, and 

 formula (55) becomes 



T t = ST - k sin 2 rs cos 2 a, 



showing that TO, = ts y when a is either 90 or 270. But when o 

 is or 180, we have 



w 1 = w-A;sm 2 w = 54 2', 



which is the minimum value of the polarizing angle. 



3. For the natural fracture-faces of the crystal the value of 

 A is 45 23'. Hence, when a = or 180, 



CT! = w - k (sin 2 w - sin 3 A) - 57 26' ; 

 and when a = 90 or 270, 



OTI = CT + k cos 2 CT sin 2 A = 59 50'. 



These values of the polarizing angles agree very well with the 

 experiments of Sir David Brewster, and still better with those 

 of M. Seebeck. 



If we wish to know in what azimuths rsi is equal to OT, on a 

 given surface of the crystal, it is obvious from (55) that we 

 must make 



sin 2 37 - sin 2 A = sin 2 w cos 2 A sin 2 a, 



whence we have, simply, 



tan A 



cosa = - , (59) 



tanw 



which shows that the thing is impossible when A is greater 

 than OT ; and that, when A is less than w, there are four such 

 azimuths ; as indeed there are, generally speaking, four values 

 of a corresponding to any other particular value of the polariz- 

 ing angle. If a' be the least of these azimuths, the others will 

 be 180- a', 180+ a', and 360- a'. On a natural face of the 

 crystal, the value of a', answering to the supposition w, = za, is 

 found to be 52 22'. 



Next, let us trace the changes which the deviation under- 

 goes in some remarkable cases. 



