Reflexion and Refraction. 117 



we shall find 



wi = -& - k cos 2 ^(sin 2 p - sin 2 2 ), (53) 



in which us is the angle whose cotangent is equal to b ; in other 

 words, 73 is the polarizing angle of an ordinary medium whose 

 refractive index is equal to the ordinary index of the crystal. 



This result accounts for a remarkable fact observed by Sir 

 David Brewster, who, in the year 1819, led the way in the ex- 

 perimental investigation of the laws of crystalline reflexion. He 

 found that the polarizing angle remains the same when the crystal 

 is turned round through 180, though one of the angles of refrac- 

 tion is changed, and though the situation of the refracted rays, 

 with respect to the axis of the crystal, becomes quite different 

 from what it was. This circumstance, which surprised me when 

 1 first met with it, is an immediate consequence of formula (53) ; 

 for the effect of a semi-revolution of the crystal is to change the 

 signs of p and q ; but the nature of the formula is such that 

 these changes of sign do not alter the value of CTL Neither is 

 that value altered by turning the crystal until the azimuth, as 

 the spherical angle A Yi is usually called, is changed into its 

 supplement ; for then the sign of p alone is affected. 



Another remark, made by the same distinguished observer, 

 is also a consequence of formula (53). From his experiments 

 it appears that, on a given surface of the crystal, the polarizing 

 angle differs from a constant angle by a quantity proportional 

 to the square of the sine of the azimuth AYi. Now, calling 

 this azimuth a, and putting A for the acute angle at which the 

 axis of the crystal is inclined to its surface, so that A may be the 

 complement of the arc A Y, we have 



sin q = cos A sin a, tan^? = cotan A cos a ; (54) 



and by making these substitutions in formula (53), after having 

 changed sin < 2 into cos w, that formula becomes 



CTI = 73 - k (sin 2 ?? - sin 2 A) + k sin 2 w cos 2 A sin 2 a, (55) 



which agrees with the remark of Brewster. 



