COLOURS OF THIN PLATES. 139 



same time in the two media ; and, consequently, that the 

 interval of retardation is the time of describing op + pr. 

 Now pr = op cos 2opq, and therefore op+pr = op (1 + cos 2opq) 

 = 2op cos 2 opq. But op cos opq =pq ; and, consequently, the 

 interval is 2pq cos opq. Or, if we denote that interval 

 by 8 ; the thickness of the plate pq, by t ; and the angle 

 opq by 0, 



$ = 2t cos 9. 



The two waves are in complete accordance or discordance 

 when the interval of retardation is an exact multiple of the 

 length of half a wave : i. e. when 



n being any number of the natural series. Equating these 

 values of S, therefore, we have, for the values of the thick- 

 ness of the plate which will produce a complete accordance 

 or discordance of the two waves, 



t = n\ sec 0. 



We learn then, 1st, that the successive thicknesses of the 

 plate, for which the intensity of the reflected light is greatest 

 or least, are as the numbers of the natural series ; 2ndly, that 

 for different species of simple light these thicknesses are 

 proportional to the lengths of the waves ; 3rdly, that for dif- 

 ferent obliquities they vary as the secant of the angle of in- 

 cidence on the exterior medium ; and, 4thly, that for plates 

 of different substances they are proportional to X, and therer 

 fore in the direct ratio of the velocities of propagation, or in 

 the inverse ratio of the refractive indices of the substances of 

 which the plate is composed. 



(153) There is one part of the preceding explanation 

 which demands a little further consideration. The two waves 



