198 Lord Rayleigh on the Manufacture and 



be composed of elements of continuously varying wave-length, 

 another factor, -, must be introduced, due to the varying elon- 

 gation of the spectra. The deficiency of light accompanying 

 an increased dispersion, however/is nothing peculiar to gratings, 

 and may be met by a proportional widening of the slit, without 

 lowering the original standard of purity. 



Another particular case of interest is obtained by consider- 

 ing a small relatively to a + d. Unless the spectrum be of high 

 order, we have simply 



B™ : B 



\a + d) ' 



so that the brightness of all the (lateral) spectra is the same. 

 According to this, the spectra of a sodium-flame observed through 

 a grating should be of equal brilliancy when a is small relatively 

 to a-\-d. In the grating with 3000 lines this condition is ful- 

 filled ; but the theoretical result is contradicted by observation, 

 the second and third lateral spectra being much brighter than 

 the first. I am not in a position to explain the discrepancy, 

 which I only noticed while drawing up this paper for the press. 

 Unless due to some mathematical blundeiy the cause would ap- 

 pear to be deep-seated. The effect of an insufficient narrowness 

 of grooves would be in the opposite direction. 



Our expressions have been obtained without taking into ac- 

 count the reflection from the face of the grating ; and therefore 

 the light stopped by the opaque parts, together with that distri- 

 buted in the central image and lateral spectra, ought to make 

 up the original brightness. Thus, if a — d, we ought to have 



1_1 %( 1 JL_ \ 



2~4~ 1 ~ 7T 2 \ 9 + 25~ f '")' 

 which is true by a known theorem. In the general case, 

 a _( a V 2 v Wi==Q0 J L . 9 fmira\ 

 a~Vd-\a + d) ^^^=1 m 2 sm { a + d J> 



a result which may be verified by Fourier's theorem. 



- By Babinet's principle it is a matter of indifference, so far as 

 the brightness of the lateral spectra are concerned, whether a 

 represent the width of the transparent or of the opaque part, 

 inasmuch as the secondary waves from the transparent and 

 opaque parts together would give a zero resultant. The same 

 conclusion may be derived from the expression for the ratio 

 B m :B. 



From the value of B m : B we see that no lateral spectrum can 

 surpass the central image in brightness ; but this result depends 

 upon the hypothesis that the lines of the grating act by opacity, 



