194 Lord Rayleigh on the Manufacture and 



examining the question for myself, I came to the conclusion 

 that the theory of gratings, as usually presented, is encumbered 

 with a good deal that may properly be regarded as extraneous. 



One of the first things to be noticed is the extraordinary pre- 

 cision required in the ruling. The difference of wave-length of 

 the two sodium-lines is about a thousandth part. If, therefore, 

 we suppose that one grating has 1000 lines in the space where 

 another has 1001, it is evident that the first grating would pro- 

 duce the same deviation for the less-refrangible D line that the 

 second would produce for the more-refrangible D line. We 

 have only to suppose the two combined into one in order to 

 see that, in a grating required to resolve the D line, there must 

 be no systematic irregularity to the extent of a thousandth part 

 of the small interval. Single lines may, of course, be out of 

 position to a much larger amount. It is easy to see, too, that 

 the same accuracy is required, whatever be the order of the spec- 

 trum examined. 



The precision of ruling actually attained in gratings is very 

 great. In the 3000 Nobert it is certain that the average interval 

 between the lines does not vary by a six-thousandth part in 

 passing from one half of the grating to the other ; for the D 

 lines, when well defined, do not appear so broad as a sixth part 

 of the space separating them. 



In considering the influence of the number of lines {n) and 

 the order of the spectrum (m), we will suppose that the ruling 

 is accurate, and that plane waves are incident perpendicularly 

 upon the face of the grating whose width is represented in the 

 figure by A B. But inas- 

 much as a large part of the 

 phenomenon covered by 

 the usual mathematical in- 

 vestigation depends upon 

 the limitation of the gra- 

 ting at A and B, we shall 

 find it convenient to take 

 first the simple case of an 

 aperture represented by 

 AB, and afterwards to 

 consider the influence of the ruling. 



In the perpendicular direction B C all the secondary waves 

 emanating from AB are in complete agreement of phase, and 

 their resultant accordingly attains its highest possible value. In 

 a direction B P, making with B C a very small angle, the agree- 

 ment of phase will be disturbed. If B P be so drawn that the 

 projection of AB upon it is equal to A., the phases of the secon- 

 dary waves will be distributed uniformly over a complete period, 



