268 



actual intensity of the reflected beam is the sum of these in- 

 tensities. Its value is found to be expressed by the formula 



_ 2 u- -{-2 i/ If' cos a +u"^ . 2 tv^-^^tvii/ cosa + w'^ 

 ^-'^^^ y l^2uu'cosa-\-u-u'- "^ ^'""^ I -^2 low' cos a +w%i^'' 



in which u and u' denote the ratios of the reflected to the in- 

 cident vibration at the two surfaces of the plate, when the 

 light is polarized in the plane of incidence ; w and iv' the 

 corresponding quantities for light polarized in the perpen- 

 dicular plane ; and a the diff^erence of phase of the successive 

 portions of the reflected beam. The values of ?<, u', to', w;, are, 



_ sw(9-e') ,_ s\n{e'-6") _ tan(e-0O ,_ tan(e-r) 

 "^-sw.d + O'Y ''-sin{d' + d"y'"~ tan(6/-l-6/V"'~tan(0'+r) 



where denotes the angle of incidence on the first surface of 

 the plate ; 0' the corresponding angle of refraction, or the 

 angle of incidence on the second surface; and 0" the angle 

 of refraction at the second. The value of a is 



a = — T cost;'; 



T being the thickness of the plate, and A the length of the 

 wave. 



When the obliquity of the incident pencil is not very 

 great, the squares and higher powers of w, ?/, ic, iv', may be 

 neglected in comparison with unity, and the expression of 

 the intensity has the approximate value, 



1-=. co%^'y(ti'^ -\- 2uu'cosa + m'") + s'm^y{tv^ + 2wtv'cosa -f tv'-) 

 This quantity will be independent of the phase a, and there- 

 fore the intensity will be constant, and the rings disappear, 



when 



ui(' COS' y + iviv'&\n-y — 0; 



tliat is, when the azimuth of the plane of polaiization has the 

 value given by the formula, 



, _ _ «^ _ _ cos {6 -d') COS (9' -e") 

 *''*""^ ~ WW' ~ cos [6 + 0) cos {0' + d ')' 



In this formula cos[9 — 6') and cos{B'—6'') are always positive ; 



