Interference-Bands. 195 



finally as the approximate value of (28) 



If now the circnmstances be such that n periods of the fi Q 

 system correspond to n — \ of the /jl system, 



1 _ (3/x -A) (8p)* 



n~ 2^^0-A) 2 ' (30 ^ 



in which the ratio of (3p — A) to 2fi does not differ much from 

 unity. In the application to extra-dense flint the simplified 

 formula 



n=(ft -A)*/(fj,-fi o y (31) 



gives very nearly the same result as that previously found. 

 The number of bands which approximately coincide is 

 inversely as the square of the range of refrangibility included. 



It must not be overlooked that the preceding investigation, 

 though satisfactory so far as it goes, is somewhat special on 

 account of the assumption that the angle of incidence (a) 

 upon the plate of air is the same for the various colours. If 

 the rays are parallel before they fall upon the prism, they 

 cannot remain parallel unless the incidence upon the first 

 surface be perpendicular. There is no reason why this should 

 not be the case ; but it is tantamount to a restriction upon 

 the angle of the prism, since a is determined by the achro- 

 matic condition. If the angle of the prism be other than 

 a, the required condition will be influenced by the separation 

 of the colours upon first entering the glass. Although the 

 general character of the phenomenon is not changed, it may 

 be well to give the calculation applicable to all angles of 

 prism, as was first done by M. Mascart. 



Denoting, as before, by «, ex! the angles of incidence and 

 refraction upon the plate of air, let /3', /3 be the angles of 

 incidence and refraction at the first surface of the prism 

 (fig. 2), whose angle is A. Then, if A, equal to nX, be the 

 retardation, 



A = nX=2tcos ot r , (32) 



as before ; while the relations among the angular quantities 

 are: — 



sin tx' = /uL sin a, ...... (33) 



a + /3 = A, (34) 



sin £' = //, sin £ (35) 



