Lord Rayleigh's Investigations in Optics. 265 



scope, when the beam is limited by prisms or gratings rather 

 than by the object-glasses of the telescopes. 



Supposing, for convenience, that the sides of the rectangle 

 are horizontal and vertical, let the horizontal aperture be a and 

 the vertical aperture be b. As in § 1, there will be no direct 

 antagonism among the phases of the secondary waves issuing 

 in an oblique horizontal direction, until the obliquity is such 

 that the projection of the horizontal aperture a is equal to JX. 

 At an obliquity twice as great the phases range over a com- 

 plete period; and, since all parts of the horizontal aperture have 

 an equal importance, there is in this direction a complete ab- 

 sence of illumination. In like manner, a zero of illumination 

 occurs in every horizontal direction upon which the projection 

 of a amounts to an exact multiple of X. 



The complete solution of the present problem, applicable to 

 all oblique directions, is given in Airy's ' Tracts/ 4th edition, 

 p. 316, and in Verdet's Lecons, t. i. p. 265. If the focal length 

 of the lens which receives 'the beam be /, the illumination I 2 

 at a point in the focal plane whose horizontal and vertical co- 

 ordinates (measured from the focal point) are (•, y, is given by 



. „ Tra P . „ 7rbr} 



T2_ «5l V. V (1) 



" X 2 f 2 7rVf 2 ttW ; ' * W 



X 2 f 2 X 2 f 2 



the intensity of the incident light being unity. The image is 

 traversed by straight vertical and horizontal lines of darkness, 

 whose equations are respectively 



sm-^=0, sm^^O (2) 



The calculation of the image due to a luminous line (of 

 uniform intensity) is facilitated in the present case by the 

 fact that the law of distribution of brightness, as one coordi- 

 nate varies, is independent of the value of the other coordinate. 

 Thus the distribution of brightness in the image of a vertical 

 line is given by 



. 9 7TClP 



+00 o 7 snr— -*■ 



m »=§4z£> • • • • (3) 



the same law as obtains for a luminous point when horizontal 

 directions are alone considered. It follows from (3) that in 

 the spectroscope the definition is independent of the vertical 

 aperture. 



i 



