REFLEXION OF POLARIZED LIGHT. 255 



have attempted in any published work to remove the 

 difficuUy, and it is not to be supposed that they had 

 despised it. 



And by the analogy of certain geometrical cases where the multi- 

 plication by -v/ — 1 indicates a line differing in angular position by 90°, 

 he hazarded the inference that such an interpretation might hold 

 good here, and that this expression would be equivalent to one of the 



form, 



27r . 2/r 



cos d sin -TT- (vl — x) + sm d sui — {vi—x + 90°; 



A A 



which is trigonometrically the same as 



2n- , \ 



( 21" , 



1 



This applying to the component in the plane of incidence, a similar 



expression would apply to that perpendicular to it, 



. / 27r \ 



or sm ( — - (vt — x) + dl ) 



The difference of these expressions, or the relative retardation of 

 the two sets of waves, will he d — 6' = 6- 



In general, 6 having any value, and the plane of polarization being 

 inclined at an angle a to the plane of incidence on the rhomb, the 

 components are, 



y = sin a sin — {vt — x + 6) (■*••) 



A 



z = cos a sin — (vt — x) (*•) 



A ^ ' 



This then is precisely the same case as that considered in a former 



note; and exactly in the same way we obtain, 



ifl 22 itiz cos 6 . „ „ 



~~-\ 5 : — = sm 2(5. 



sm 2« cos ^a cos a sm a 



The general equation to an ellipse. If d = 90°, the semi-axes are 

 sin a and cos a, parallel and perpendicular to the plane of incidence. 

 If a =^ 45° and 6 variable, it is still an ellipse. If a = 45° and 6 = 90°, 

 it becomes a circle. Tims a ray polarized at an angle a, loith th^ plane 

 of incidence, after two internal reflexions in glass, emerges elliptically or 

 circularly polarized, according to the above condition. 



From the empirical terms before mentioned, Fresnel derived ex- 

 pressions from which he calculated that for crown glass, where (i = 

 1'51, an internal incidence i = 54° 37' would give 6 = 45°. Thus 

 experimentally cutting a rhomb of such glass at that angle, so that 

 the ray polarized at 45° to the plane of incidence, entering one face 



