AND TRANSMITTED BY THIN PLATES. 159 



= 2 r cos 6', T being the thickness of the plate, and & the angle of 

 refraction ; and the corresponding difference of phase is 



4"" n, 



a = -r- T cos a ; 

 A 



A being the length of an undulation. The difference of phase, 

 after three internal reflexions, is 2a ; after five internal reflexions, 

 3a ; and so on. Hence, if ^ denote the phase of the vibration 

 reflected at the first surface, at the instant of reflexion, - a will 

 be the phase of the portion which emerges there after one internal 

 reflexion ; $ - 2a, after three, &c. And the sum of all the inter- 

 nally-reflected vibrations will be 



u-i (1 - u~) [sin (< - a) - uu 2 sin ($ - 2a) + u % u sin (0 - 3a) - &c.] . 

 in which, it can be easily shown, the quantity within the brackets 

 is equal to 



sin (0 - a) + uu z sin <j> 



1 + 2ww 2 cos a + w 2 M 2 2 * 



Adding u sin $, the vibration reflected externally at the first 

 surface of the plate, the sum of all the reflected vibrations is 



,. sin (0 - a) + uu-i sin 6 



u sin d> + 2 (1 - w 2 ) ^ - ,-V 



' 1 + 2uu 2 cos a + wW 



Let this quantity be put under the form 



P sin + Q cos ^. 

 Then we find 



cos o + uu t 



. 



P = U + U 2 (1 - U) 



; 1 + 2uui COS a + W 



- w 2 (1 - w 2 ) sin o 

 1 + 2miz cos a + ifu*' 



IVherefore, the intensity of the resulting light is* 



t/ 2 + 2m/ 2 cos a +t/^ t 

 * "l+2tt,COSa + V 



,nd its phase, ;//, will be given by the formula, 

 -_Q = M 2 (1 - M 2 ) sin a 



n ^ ~ P ~ u (1 + w./j + ih (1 + ") cos a' 



* This expression for the intensity has been alieady obtained by Mr. Airy "On 

 the Phenomena of Newton's Rings, when formed between two transparent substances 

 of different refractive powers." Camb. Tram., 1832. 



