204 Lord lxayleigh on Copying Dij/'raction-gnitings, 



+ A, cos f—j- + c 2 ) cos(/cz—/u, 2 z) + ] a 



+ [— -Aj cos (^- + eA sin (fcz—fj, it) 

 + B t cos ( :1 y- + e' J cos(/ft8f — fi x z) 



— A 2 cos( :L 7 1 -t-e 2 )sm{fcz — fi z z) -f ] 2 . 



In order to apply this result to our present question, it is 

 supposed as a rough approximation that the terms with suf- 

 fixes higher than one may be omitted. We thus obtain 



A 2 + JA^ + JBj 2 + 2A Aj cos (-^ + e^ cos (kz—^z) 

 — -j- + <?/ ) sinfaz—fjuz) 



which as a function of z is periodic with a period determined 

 by fcz—fA 1 z=27r, or x 



V('-S)' 



In the cases with which we are concerned X 2 is small in 

 comparison with d 2 , so that approximately z — 2d 2 /X. So far, 

 then, as this theory extends, the phenomena behind the grating- 

 are reproduced with every retreat through a distance 2d 2 /X; 

 but, on account of the terms omitted, this conclusion does not 

 apply to the subordinate periods (on which depends the per- 

 formance of a copy in the spectra of higher order) ; nor does 

 it apply rigorously even to the principal period itself. 



Similar results to those given by direct inspection on the 

 coarse grating have been obtained by photographic copying 

 of finer ones, a lens (as already explained) being substituted 

 for flat glass as a support for the sensitive film. When the 

 copy is held so that the spectrum of the first order is formed 

 upon the eye, several dark rings are visible, separated by in- 

 tervals of brightness. With the 6000 Robert the diameter 

 of the first dark ring was *54 inch, and at the centre round 

 the point of contact there was a dark spot nearly as dark as the 

 ring. In the second and third spectra the centres were also 

 dusky, though not so black as in the first. The diameter of 

 the first dark ring in the second spectrum was "30 inch. 



The occurrence of a dark centre is a point of interest, as 

 showing that for purposes of reproduction it is possible for the 



