Theory of Diffraction- gratings. 199 



which is generally very far from being true. In an engraved glass 

 grating or in a gelatine copy there is no opaque material present 

 by which light could be absorbed, and the effect depends on a 

 difference of retardation due to the alternate parts. It is re- 

 markable that this point is never alluded to in the ordinary 

 treatises on optics, and, so far as I know, was first noticed by 

 Quincke (Pogg. Ann. vol. cxxxii. p. 321 (1867)), who made a 

 theoretical and experimental examination of the phenomena pre- 

 sented when light is diffracted at the edge of a transparent 

 obstacle. My attention was first drawn to it, before I was ac- 

 quainted with Quincke's work, by observing that, contrary to 

 my anticipations, it was possible for the lateral spectra of a soda- 

 flame to exceed the central image in brilliancy. When once the 

 question is raised, the explanation is easy enough ; for if the 

 grating were composed of equal alternate parts, both alike trans- 

 parent but giving a relative retardation of half a wave-length, it 

 is evident that the central image would be entirely extinguished, 

 while the first spectrum would be four times as bright as if the 

 alternate parts were opaque. If it were possible to introduce at 

 every part of the aperture of the grating an arbitrary retardation, 

 all the light might be concentrated in any desired spectrum. By 

 supposing the retardation to vary uniformly and continuously, 

 we fall on the case of the ordinary prism ; but there would then 

 be no analysis of light, except such as depends on the variation 

 of retardation with wave-length. To obtain a diffraction-spec- 

 trum in the ordinary sense containing all the light, it would be 

 necessary that the retardation should gradually alter by a wave- 

 length in passing over each element of the grating and then fall 

 back to its previous value, thus springing suddenly over a wave- 

 length. It is not likely that such a result will ever be obtained 

 in practice ; but the case was worth stating, in order to show that 

 there is no theoretical limit to the concentration of light of 

 assigned wave-length in one spectrum, and as illustrating the 

 frequently observed unsymmetrical character of the spectra on 

 either side of the central image. 



We have now to consider the dependence of the resolving- 

 power of a grating on the number of its lines (n) and the order of 

 the spectrum observed (m). A B 



Let B P be the direction ' 

 of the principal maximum 



for the wave-length X in // 



the mth spectrum ; then /} 



the projection of AB on 

 BPiswmX. If BQ be the 

 direction corresponding to 

 the first minimum, the pro- 



