in Theory and Practice. 407 



or 



X> — 2 , 2 ? 2 5 <xo * 



, T 1 a 3 a 5 a 

 2yi 2y x 2^ 



a 



Thus when — =2, the 1st, 3rd, &c, spectra will disappear, 

 making a grating of twice the number of lines to the centim. 

 When — = 4, the 2nd, 6th, 10th, &c, spectra disappear. 



yi 



When— =6, the 3rd, 9th, &c, spectra disappear. 



The case in which— =4, as Lord Rayleigh has shown, 

 «y i 

 would be very useful, as the second spectrum disappears leaving 

 the red of the first and the ultra-violet of the third without 

 contamination by the second. In this case two lines are 

 ruled and two left out. This -would be easy to do, but the 

 advantages would hardly pay for the trouble owing to the 

 following reasons : — Suppose the machine was ruling 20,000 

 lines to the inch. Leaving out two lines and ruling two 

 would reduce the dispersion down to a grating with 5000 

 lines to the inch. Again, the above theory assumes that the 

 grooves do not overlap. Now I believe that in nearly, if not 

 all, gratings with 20,000 lines to the inch the whole surface is 

 cut away and the grooves overlap. This would cause the 

 second spectrum to appear again after all our trouble. 



Let the grooves be nearly equidistant, one being slightly 



displaced. In this case y 1 =~+v. 



COS' 77 



N#! / 7rN 77-Nv . ?tN . 7rNv\- 



— - = cos-tt-cos sin -rrsin 



2 \ 2 a 2 a J 



For the even spectra this is very nearly unity, but for the odd 

 it becomes 



«)'■ 



Hence the grating has its principal spectra like a grating of 



a 

 space - ; but there are still the intermediate spectra due to 



the space a, and of intensities depending on the squares of the 

 order of spectrum, and the squares of the relative displace- 



* A theorem of Lord Rayleigh 's. 



