84 Lord Rayleigh on Achromatic 



it. In illustration of this, two extreme cases may be con- 

 sidered of a slit illuminated by ordinary sunshine. First, let 

 the width w 2 be great enough not sensibly to dilate the solar 

 image; that is, let w 2 be much greater than X/s, where s denotes 

 in circular measure the sun's apparent diameter (about 30 

 minutes). In this case the light streams through the slit 

 according to the ordinary law of shadows, and the pupil (of 

 diameter p) will be filled with light if situated at a distance 

 exceeding d*, where 



p/d = s (6) 



At this distance the apparent width of the slit is iv 2 /d, or w^s/p; 

 and the question arises whether it lies above or below the 

 ocular limit X/p, that is, the smallest angular distance that 

 can be resolved by an aperture p. The answer is in the 

 affirmative, because we have already supposed that w 2 s exceeds 

 X. The slit has thus a visible width, and it is seen backed by 

 undiffracted sunshine. If a spectrum be now formed by the 

 use of dispersion sufficient to give a prescribed degree of 

 purity, it is as bright as is possible with the sun as ultimate 

 source, and would be no brighter even were the solar diameter 

 increased, as it could in effect be by the use of a burning- 

 glass throwing a solar image upon the slit. The employment 

 of a telescope in the formation of the spectrum gives no 

 means of escape from this conclusion. The precise definition 

 of the brightness of any part of the resulting spectrum would 

 give opportunity for a good deal of discussion ; but for the 

 present purpose it may suffice to suppose that, if the spectrum 

 is to be divided into n distinguishable parts, so that its angular 

 width is n times the angular width of the slit, the apparent 

 brightness is of order 1/n as compared with that of the sun. 



Under the conditions above supposed the angular width 

 of the slit is in excess of the ocular limit, and the distance 

 might be increased beyond d without prejudice to the brilliancy 

 of the spectrum. As the angular width decreases, so does the 

 angular dispersion necessary to attain a given degree of 

 purity. But this process must not be continued to the point 

 where wjd approaches the ocular limit. Beyond that limit 

 it is evident that no accession of purity would attend an in- 

 crease in d under given dispersion. Accordingly the dis- 

 persion could not be reduced, if the purity is to be maintained; 

 and the brightness necessarily suffers. It must always be a 

 condition of full brightness that the angular width of the slit 

 exceed the ocular limit. 



Let us now suppose, on the other hand, that w 2 is so small 



* About 30 inches. 



