8o On the Laws of Reflexion 



the polarizing angles in various azimuths agree very well with 

 your experiments. You will perceive that the value of 8 is the 

 same in supplementary azimuths, which explains the observation, 

 cited in the beginning of my letter, relative to the equality of 

 the polarizing angles at opposite sides of the perpendicular IZ 

 in a given plane of incidence. 



When the point JR falls upon 0, we have 8 = 0, and i + <j> 

 equal to a right angle. Hence, when the cotangent of ZR is 

 equal to the ordinary index, the tangent of the polarizing angle 

 is equal to the same index. This theorem, though deduced from 

 an approximate equation, might be shown to be exact. 



When the axis of the crystal lies in the plane of incidence, we 

 may obtain an exact expression for the polarizing angle. The 

 condition of polarization then becomes 



cos ( + f) - (< - V) sin f cos f . '* - ; (6) 



bill \^i ({) J 



from which, by the proper substitutions, we obtain the following 



expression : 



. ' . 1 - a 2 cos 2 A - b 2 sin 2 A 



sm 2 * = -- - ; (7) 



i - a 2 b 2 



where A denotes the complement of ZP, or the inclination of the 

 axis to the face of the crystal, and i is the polarizing angle. 

 This formula, in a shape somewhat different, was communicated, 

 above a year ago, to Professor Lloyd, who has noticed it, in con- 

 nexion with your Paper, in his " Eeport on Physical Optics." 

 When a and b become equal, the formula gives your law of the 

 tangent for ordinary media. 



The foregoing results show that, when a ray is polarized by 

 reflexion from a crystal, the plane of polarization deviates from 

 the plane of incidence, except when the axis lies in the latter 

 plane ; and that the deviation may be made very great by 

 placing the crystal in contact with a medium whose refractive 

 power is nearly equal to that of the crystal itself ; for when i is 

 nearly equal to or to $', the divisor sin (i - <j>) or sin (i - $') is 



