Interference-Bands. 191 



relative retardation of the two pencils at the point u is 



and the situation of the central, or achromatic, band is de- 

 termined, not by 4> = 0, but by d<f>/d\ = 0, or 



w = \ 2 DF / (\)/^ ...... (22) 



It is scarcely necessary to say that although the nth band 

 may be rendered achromatic, the system is no more achromatic 

 than if the prism had been dispensed with. The width of the 

 component systems being unaltered, the manner of overlapping 

 remains as before. The present use of the prism is of course 

 entirely different from that previously discussed, in which 

 by a suitable adjustment the system of bands could be 

 achromatized. 



Thin Plates. 



The series of tints obtained by nearly perpendicular re- 

 flexion from thin plates of varying thickness is the same as 

 that which occurs in Lloyd's interference experiment, or at 

 least it would be the same if the material of the plates were 

 non-dispersive and the reflecting power small. If t be the 

 thickness, fi the index, ex! the inclination of the ray within 

 the plate to the normal, the relative retardation of the two 

 rays (reckoned as a distance) is 2 fit cos ex! , and is sensibly 

 independent of X. 



" This state of things may be greatly departed from 

 when the thin plate is rarer than its surroundings, and 

 the incidence is such that ex! is nearly equal to 90° ; for 

 then, in consequence of the powerful dispersion, cos ex! may 

 vary greatly as we pass from one colour to another. Under 

 these circumstances the series of colours entirely alters its 

 character, and the bands (corresponding to a graduated 

 thickness) may even lose their coloration, becoming sensibly 

 black and white through many alternations f . The general 

 explanation of this remarkable phenomenon was suggested by 

 Newton, but it does not appear to have been followed out in 

 accordance with the wave theory. 



" Let us suppose that plane waves of white light travelling 

 in glass are incident at angle ex. upon a plate of air, which is 

 bounded again on the other side by glass. If /n be the index 

 of the glass, a! the angle of refraction, then sin a! = fi sin. ex ; 

 and the retardation expressed by the equivalent distance in 

 air > 1S 2t sec a! —fju 2t tan ex! sin oc = 2t cos ex' ; 



* Enc. Brit. Wave-Theory, vol. xxiv. p. 425. 



t Newton's Optics, Book ii. ; Fox Talbot, Phil. Mag. ix. p. 401 (1836). 



