88 Lord Rayleigh on Achromatic 



practice that, to reproduce a grating by photography, it is 

 necessary that the sensitive plate be brought into close contact 

 with the original ; whereas, according to the argument just 

 advanced, no such limitation would be required. 



These discrepancies will be explained if, starting from the 

 general theory, we take into account the actual constitution of 

 the gratings with which we can experiment. If plane waves 

 of homogeneous light (X) impinge perpendicularly upon a 

 plane (e = 0) grating, whose constitution is periodic with 

 respect to x in the interval cr, the waves behind have the 

 general expression 



A cos (kat — kz) +A 1 cos (_p#+/i) cos {kat— \x x z) 

 + Bi cos (px + gi) sin [kat — ^z) 

 + A 2 cos {2px+f 2 ) cos (kat—fjL 2 z) + . . . ; . (8) 



where 

 and 



P = 27t/<t, k = 27r/X, 

 fi 1 2 = k 2 —p 2 , /A,2 2 = k 2 — 4:p 2 , &c.j 



the series being continued as long as //, is real*. Features in 

 the wave-form for which jjl is imaginary are rapidly elimi- 

 nated. For the present purpose we may limit our attention 

 to the first three terms of the series, which represent the 

 central image and the two lateral spectra of the first order. 



When the first term occurs, as usually happens, the phe- 

 nomena are complicated by the interaction of this term with 

 the following ones, and the effect varies with z in a manner 

 dependent upon \. This is the ordinary case of photographic 

 reproduction, considered in the paper referred to. If A 

 vanish, there is no central image ; but various cases may still 

 be distinguished according to the mutual relations of the other 

 constants. If only A ]? or only B 1? occur, we have interference- 

 fringes. The intensity of light is (in the first case) 



A 1 s cos s .fcp«+/), (9) 



vanishing when . . , , tA 



& px+f=\(n + l)TT \ 



and these fringes may be regarded as arising from the inter- 

 ference of the two lateral spectra of the first order, 



^Aj cos (kat—fjbiz+px +/i)> 

 \K X cos (kat—foz —px — _/i) . 

 As an example of only one spectrum, we may suppose 



* Phil. Mag. March 1881 ; Enc, Brit. Wave Theory, p. 440. 



