158 LORD RAYLEIGH ON THE COLOURS OF THIN PLATES. 



The complementary colours, best obtained with the aid of polarised light, 

 are also calculated and exhibited on a diagram. 



§ 1. The calculation, according to Young and Poisson, of the amount of 

 light of given wave-length (X) reflected from a thin plate is given in all treatises 

 on physical optics. If D be the thickness, /3 the obliquity of the ray within 

 the plate, 1 : e, the ratio in which the amplitude is altered in one reflection, 

 then for the intensity of light in the reflected system we have— 



4^sin2(7rV/X) -. 



(l-e 2 ) 2 + 4e 2 sin 2 (7rV/A) { } 



in which the intensity of the original light is taken to be unity, and V is 

 written for 2 D cos /3. The colours exhibited in white light are to be found 

 by combining the chromatic effects of all the rays of the spectrum. 



When, as in Newton's rings, the thickness of the plate varies from point to 

 point, there is a series of colours determined by supposing D to vary in the 

 above expression. This series is not absolutely independent of the material of 

 which the plate is composed, even if we disregard the differences of brightness 

 corresponding to the occurrence of e 2 in the numerator of our expression. On 

 account of retarded propagation, the value of X for a given ray is less in glass, 

 for instance, than in air ; and in consequence of dispersion there is no accurate 

 proportionality, so that we cannot say absolutely that a definite thickness in 

 glass corresponds to a definite, though different, thickness in air. Moreover, 

 since e varies from one body to another, the denominator of (1) changes its 

 value somewhat. 



It is evidently impracticable to carry out calculations strictly applicable to 

 all cases. If we take for X the wave-length in air, we obtain results appropriate 

 to the ordinary case of Newton's rings ; and in extending them to plates of 

 other material, we in effect neglect the relatively small influence of dispersion. 



Again, we may without much error neglect the variation of the denomin- 

 ator with wave-length, which amounts to supposing e 2 small, or that the two 

 media do not differ much in refrangibility. In the case of glass and air the 

 value of e 2 is about ^5. When sin 2 (7rV/X) is small, it is of little consequence 

 what the value of the denominator may be, and we may therefore identify it 

 with (1 + e 2 ) 2 , taking instead of (1), 



4e 2 . „ 7rV /on 



sin 2 (2) 



(1+e 2 ) 2 " X 



It is on this formula, strictly applicable only to a plate of air bounded by 

 matter of small refrangibility, that the calculations and diagrams of this 

 investigation are based. 



§ 2. The colours of Newton's scale are met with also in the light transmitted 



