8o On the Laivs 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 R falls upon 0, we have 8 = 0, and i + $ 

 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 



o 

 R1T1 4 



cos (i + 6') - (a 2 - 6 2 ) sin f cos tf -r-~ . = ; (6) 



r enn(t-f) 



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



expression : 



1 - 2 cos 2 A - 6 2 sin 2 A 



sin 2 * = - ; (7) 



i - a 2 o 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 shajie somewhat different, was communicated, 

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

 nexion with your T &p&.i n his "Report on Physical Optics." 



When a and b beco; ^jfes . aa j^ thefarSfe^F 68 70Ur 1&W f the 

 tangent for ordinary M nc J. ^*x 



The foregoing r<Jp , g ^ ^ when a ray is oolarized by 

 reflexion from a cry^ ^ plane of polarization deviates from 

 the plane of inciden^' except when ^ axis ^eg i n the latter 

 plane ; and that ^ ' deviation may fa made very great by 

 placing the crys^ in contact ^^ a me dium whose refractive 

 power is nearly eq ual ^ that of the crystal itse lf ; for when i is 

 nearly equal to <X T to ^ divifl(jr sin (,- _ fi O r sin (i - 0') is 



