Theory of DiJJr action-gratings. 197 



nation is distributed according to the same law as for the central 

 image (m = 0), vanishing, for example, when the retardation 

 amounts to (mn±l)\. In considering the relative brightness 

 of the different spectra, it is therefore sufficient to attend merely 

 to the principal directions, provided that the whole deviation be 

 not so great that its cosine differs considerably from unity*. 



Under the restriction just stated, the intensity of the secondary 

 waves may be supposed not to be diminished by the obliquity ; 

 and thus we obtain for the ratio of brightness : — 



amir 



o \_}amn a + dJ \am<ir J a + d 3 



*^ a+d 



where B m denotes the brightness of the mih. spectrum, and B 

 of the central image. 



If B denotes the brightness of the central image when the 

 whole of the space occupied by the grating is transparent, we 

 have 



B e :B = a 2 :(« + d) 2 , 

 and thus 



B,„ : B= -rr-^sin^ 



a + d 



The sine of an angle can never be greater than unity; and con- 

 sequently under the most favourable circumstances only — 3-3 of 



the original light can be obtained in the mth spectrum. We 

 conclude that, with a grating composed of transparent and opaque 

 parts, the utmost light obtainable in any one spectrum is in the 

 first (the central image not being included), and there amounts 



to — 2 , or about ^j, and that for this purpose a and d must be 



7T 



equal. 



When d=a, the general formula becomes 



showing that, when m is even, B m vanishes, and that, when m is 



odd, B m : B = ^-=- 

 nrir- 



The third spectrum has thus only one ninth of the brilliancy 



of the first. 



It is here supposed that the light is homogeneous; but if it 



* This point is perhaps made clearer by supposing the original light to 

 be always incident at such an angle that the diffracted spectrum under con- 

 sideration occurs in the normal direction. 



