166 ON THE LIGHT KEFLECTED 



Revolving this, we find 



1 + uu z cos a 



= W " 



a _ , , 



+ W' * 1 



1 + 2MW, COS a + W + 2WM 2 COS a + tt'tt, 



Hence the intensity of the transmitted light is 



r 



and the phase is given by the formula 



., Q - wwa sin o 



tan ;// = - -g = -= --- . 

 P 1 + UU Z COS a 



13. When u and u, have the same sign, the greatest and least 

 intensities correspond to a = (2m + 1) TT, and a = 2imr. They are, 

 respectively, 



, 



and 



1 - 



When u and M 2 have opposite signs, the greatest and least values of 

 /' correspond to a = 2mtr, and a = (2m + I)TT. Hence the bright 

 rings of the transmitted system occur at thicknesses which produce 

 the dark rings of the reflected system, and vice versa. 



These values never vanish, since u' and M' Z are never evanes- 

 cent ; and there are no Hack rings in the transmitted light. 



14. When the incident light is polarized perpendicularly to the 

 plane of incidence, and when the coefficients u and u 2 are both 

 positive, or both negative, the preceding values of the greatest and 

 least intensities remain unaltered. In other words, the rings 

 exhibit the same character, whether the angle of incidence is less 

 than the polarizing angle at both surfaces, or greater. But when 

 one of these quantities is positive, and the other negative, i.e. 

 when the incidence at one surface is less than its polarizing angle, 

 and at the other greater, the preceding formulas are transposed, 

 and the bright rings are changed into dark ones, and nee versa. 

 When u = 0, or u z = 0, i.e. when the light is incident on either 

 surface at its polarizing angle, the preceding values are both re- 

 duced to (wV 2 ) 2 , and there is no variation of intensity caused by 

 interference. 



15. It may be easily shown from Fresnel's formulas, that 



