300 Lord Rayleigh on the Limit to Interference 



If the two streams are obtained by reflexion at the opposite 

 faces of a parallel plate, the circumstances are somewhat more 

 complicated. But the simple theory is applicable even here 

 as a first approximation, which becomes more and more rigo- 

 rous as the difference of optical quality between the plate and 

 the medium in contact with it is supposed to diminish. If ft 

 be the index of the plate, A its thickness, 



5. 47r;uA 47rw/tiA ,„. 



x""*^ — V — ■ • • • v) 



If the plate be of air, /a = 1. In any case the variation of ft 

 is small compared to that of ?i ; so that if A denote the equi- 

 valent thickness of air, we may take 



1=4 sin^-^^, (8) 



a function of n — the frequency, as well as of A and V. 



If now the light be heterogeneous, we have nothing further 

 to do than to integrate (8) with respect to n, after introduc- 

 tion of a factor i such that i dn represents the illumination 

 corresponding to dn*. In the present case, where the inten- 

 sity is supposed to be uniform within limits Wi and n^, and to 

 vanish outside them, we have 



J I dn = 4.i j %in2 {2'j7nA/Y) dn 



From this we fall back on (8), if we suppose that (n^^ — ni) is 

 infinitely small, so that 



^Idn = 2^idn. [1— cos47rwA/V]. 



The difference between (8) and (9) thus depends upon the 

 factor 



sin27rA(yi2-ni)/V 

 2'rrA(n,-nj)/Y ' ^^^^ 



which multiplies the second term of (9). If we introduce the 

 special values of ??i, n2 from (4), and denote the angle in (10) 

 by a, 



«=27rA(«2-/ii)/V=^.^. . . (11) 



* It is here assumed that the range included is too small to give rise to 

 sensible chromatic variation. 



