Interference- Bands. 85 



that the image of the sun is dilated to many times s, or that 

 w 2 is much less than X/s. The divergence of the light is now 

 not s, but X/w 2 ; and, if the pupil be just immersed, 



p/d=X/w 2 . 



The angular width of the slit w 2 /d is thus equal to X/p, that 

 is, it coincides with the ocular limit. The resulting spectrum 

 necessarily falls short of full brightness, for it is evident that 

 further brightness would attend an augmentation of the solar 

 diameter, up to the point at which the dilatation due to 

 diffraction is no longer a sensible fraction of the whole. In 

 comparison with full brightness the actual brightness is of 

 order iv 2 s/X ; or, if the purity required is represented by n, 

 we may consider the brightness of the spectrum relatively to 

 that of the sun to be of order iv 2 s/(nX). 



In the application of these considerations to Lloyd's bands 

 we have to regard the narrow slit w% as illuminated, not by 

 the sun of diameter s, but by the much narrower source 

 allowed by the first slit, whose angular width is wjY). On 

 this account the reduction of brightness is at least ?# 1 /(sD). 

 If w x be so narrow as itself to dilate the solar image, a further 

 reduction would ensue; but this could always be avoided, 

 either by increase of D, or by the use of a burning-glass 

 focusing the sun upon the first slit. The brightness of the 

 spectrum of purity n from the second slit is thus of order 



Wi iv 2 s w } iv 2 



sD ' nX nXD ' 



We have now to introduce the limitations upon iu 1 and iv 2 . 

 By (4) iv 1 must not exceed 5/(4n); and by (2) w 2 must not 

 exceed XD/(2b) . Hence the brightness is of order 



l/(8» 2 ), (7) 



independent of s, and of the linear quantities. The fact that 

 the brightness is inversely as the square of the number of 

 bands to be rendered visible explains the somewhat sudden 

 failure observed in experiment. If w = 2000, the original 

 brightness of the sun is reduced in the spectrum some 30 

 million times, beyond which point the illumination could 

 hardly be expected to remain sufficient for vision of difficult 

 objects such as narrow bands. 



In Fresnel's arrangement, where the light is reflected per- 

 pendicularly from two slightly inclined mirrors, interference 

 of high order is obtained by the movement of one of the 

 mirrors parallel to its plane. The increase of n does not then 

 entail a narrowing of w x ; and bands of order n may be 



