268 



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

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



„ u"^ -{-2 lit/' cos a +ti'^ , . q w^ -^ 2 wti/ cos a -\-w'^ 



1 = C0S^7 -;-— ; TT^ + Sin-7 , , ^ ; ; 5— 2> 



\-{-2uucosa-\-u-u- L -\- 2 ww cos a -{-iV'w' 



in which u and u' denote the I'atios 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 w' the 

 corresponding quantities for light polarized in the perpen- 

 dicular plane ; and a the difference of phase of the successive 

 portions of the reflected beam. The values of m, u', w', vj, are, 



_ sin(0-eO ,_s\n{e^-e^ __t an{9-9') ,_ tan(0'-r) 

 ^~sm[e + e'y'^~sm{d' + 8''y'^~ tan(0+OO''"~tan(0' + r) 

 where denotes the angle of incidence on the first surface of 

 the plate ; 9' the corresponding angle of refraction, or the 

 angle of incidence on the second surface; and 9" the angle 

 of refi'action at the second. The value of a is 



a= -r-T cos^'; 

 A 



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

 wave. 



When the obliquity of the incident pencil is not very 

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

 neglected in comparison with unity, and the expression of 

 the intensity has the approximate value, 



1=: cos'^'y(M^ -t" 2tm'cosa + u'"^) +'s\n^y{w^ -f- 2wtv'cos a + tv'-) 

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

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



when 



tin' cos^y + ^Vlv's'm-y — 0; 



that is, when the azimuth of the plane of polarization has the 

 value given by the formula, 



- - _ 'iill - _ cos (0-00 cos (y-r) 

 ^'*"' ^ ~ WW' ~ cos {9 + 9') cos {9' +9')' 



In this formula cos{9 — 9') and cos(0'—0") are always positive; 



