198 Lord Rayleigh on Achromatic 



formed a little beyond the limit of total reflexion. They are 

 broad and richly coloured if the layer of air be thin, but as 

 the thickness increases they become finer, and the colour is 

 less evident. 



The theoretical condition of constant thickness is better 

 satisfied if (after Mascart) we place the layer of air in the 

 focus of a small radiant point (e.g. the electric arc) as formed 

 by an achromatic lens of wide angle. In this case the 

 area concerned may be made so small that the thickness in 

 operation can scarcely vary, and the ideal Herschel's bands 

 are seen depicted upon a screen held in the path of the re- 

 flected light. It will of course be understood that bands 

 may be observed of an intermediate character in the formation 

 of which both thickness and incidence vary. Herschel's 

 observations relate to one particular case — that of constant 

 thickness; Talbot's to the other especially simple case of 

 constant angle of incidence. 



From our present point of view there is, however, one very 

 important distinction between the two systems of bands. The 

 one system is achromatic, and the other is not. In order to 

 understand this, it is necessary to follow in greater detail the 

 theory of Herschel's bands. 



We will commence by supposing that the light is homo- 

 geneous (X constant), and inquire into the law of formation of 

 the bands, t being given. The same equations, (32) &c, 

 apply as before, and also fig. 2, if we suppose the course of 

 the rays reversed, so that the direction of the emergent ray is 

 determined by 0'. The question to be investigated is the 

 relation of (S f to n, and to the other data of the experiment. 



The band of zero order (n = 0) occurs when of = 90°, that is 

 at the limit of total reflexion. The corresponding values of 

 a, /3, and ft may be determined in succession from (33), (34), 

 (35). The value of a! for the nth band is given immediately 

 by (32). For the width of the band, corresponding to the 

 change of n into n + 1, we have 



X= — 2£smVda', 



and from the other equations, 



cos a! da! = fM cos a da, 



da + d/3 = 0, 



cos ff dff = fi cos /3 dfi ; 



so that the apparent width of the nth. band is given by 



d0 = % j*** . , (37) 



4r cos p cos a sin a' v ' 



In the neighbourhood of the limit of total reflexion sin a! 



