The TJicory of Optical Images. By Lord, Raylcigh. 471 



"When A r has appeared, its value is proportional to the ordinate 

 at x = s. Thus in the case of a circular aperture (a = 2 E) we 

 have 



^ = ,= EV{1 -r 2 \ 2 /e 2 sin 2 a}. . . . (63) 



The above investigation relates to a row of luminous points 

 emitting light of the same intensity and phase, and it is limited 

 to those points of the image for which t] (and q) vanish. If the 

 object be a grating radiating under similar conditions, we have 

 to retain cos q y in (12) and to make an integration with respect 

 to q. Taking this first, and introducing a factor c ±k i, we have 



+ a> *.*. , 2 h 



i 



e^^cofiqydq — ^—-,. . . . (64) 



This is now to be integrated with respect to y between the 

 limits — y and -f- y. If this range be finite, we have 



Limit k = 72 ■ 2 ~ 2 ^» • • • C 65 ) 



J - y /c + y 



independent of the length of the particular ordinate. Thus 



;+ =» 

 'G dq as 27r fcospx dx, . . (66) 

 — 00 



the integration with respect to x extending over the range for 

 which y is finite, that is, over the width of the object-glass. If 

 this be 2 E, we have 



8 e<fy = 47r/p.sinpE. . . (67) 



/:: 



From (67) we see that the image of a luminous line, all parts 

 of which radiate in the same phase, is independent of the form of 

 the aperture of the object-glass, being, for example, the same for 

 a circular aperture as for a rectangular aperture of equal width. 

 This case differs from that of a self-luminous line, the images of 

 which thrown by circular and rectangular apertures are of different 

 types.* 



The comparison of (67) with (20), applicable to a circular 

 aperture, leads to a theorem in Bessel's functions. For, when q is 

 finite, 



s/.W + f) ' ■ ■ ( } 



so that, setting E = 1, we get 



V(p 2 + r) v 



r 



* Enc. Brit., 'Wave Theory,' p. 431. 



+ This raiiy be verified by means of Neumann's formula (Gray and Mathews, 

 4 Bessel'B Functions' (70) p. 27). 



