Crystalline Reflexion and Refraction. 57 



where 



(a«_i2)(l+i2)_ 



(51) 



^ 



this value of K being found by assuming tan tj=:cotan 4^=6, which is accurate 

 enough for the purpose. 



Thus we have obtained i^-\- 1,^, or the sura of the polarising angle and the 

 angle of ordinary refraction. The former angle itself may be inferred from 

 formula (50) by help of the relation sin/j=:isintj. In this way, if we use w, 

 instead of <, to distinguish the polarising angle from other angles of incidence, 

 and if we put 



we shall find 



■isr^:=:-sy—kcos-q{s\n''p — sin^f^^), (53) 



in which -ar is the angle whose cotangent is equal to b ; in other words, w is the 

 polarising 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 experimental investigation of the 

 laws of crystalline reflexion. He found that the polarising angle remains the 

 same when the crystal is turned round through 180°, though one of the angles 

 of refraction 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 I first met with it, is an immediate con- 

 sequence of formula (53) ; for the effect of a semi-revolution of the crystal is to 

 change the signs of^ and q ; but the nature of the formula is such that these 

 changes of sign do not alter the value of ra-,. Neither is that value altered by 

 turning the crystal until the azimuth, as the spherical angle AY« is usually 

 called, is changed into its supplement ; for then the sign of j9 alone is affected. 



Another remark, made by the same distinguished observer, is also a conse- 

 quence of formula (53). From his experiments it appears that, on a given 

 surface of the crystal, the polarising angle differs from a constant angle by a 

 quantity proportional to the square of the sine of the azimuth AYi. Now 



VOL. xvni. / / ^ 



