The Theory of Optical Images. By Lord Rayleigh. 461 



The illumination at various parts of the image of a double point 

 may be investigated as before, especially if we limit ourselves to 

 points which lie upon the line joining the two geometrical images. 

 The only difference in the calculations is that represented by the 

 substitution of 2 J! for sine. We shall not, however, occupy space 

 by tables and drawings such as have been givenjfor a rectangular 

 aperture. It may suffice to consider the three principal points in 

 the image due to a double source whose geometrical images are 

 situated at u = and u = — ir, these being the points just men- 

 tioned and that midway between them at u = — £7**. The values 

 ■of the functions required are 



2J 1 (0)/0 = 1*0000 = Vfl'OOOO}. 



2J 1 (tt)/w = "1812 = J\ [-03283}. 



2Ji(K)/-|7r = '7217 = J{> 5209}. 



In the case (corresponding to i. fig 4) where there is similarity 

 •of phase, we have at the geometrical images amplitudes 1*1812 as 

 against 1*4434 at the point midway between. When there is 

 ■opposition of phase the first becomes + *8188, and the last zero.* 

 When the phases differ by a quarter period, or when the sources 

 are self-luminous (iii. fig. 4), the amplitudes at the geometrical 

 images are *J{ 1*0328} or 1*0163, and at the middle point 

 J{ 1 • 0418 } or 1 * 0207. The partial separation, indicated by the 

 •central depression in curve iii. fig. 4, is thus lost when the rect- 

 angular aperture is exchanged for a circular one of equal width. 

 It should be borne in mind that these results do not apply to a 

 double line, which in the case of a circular aperture behaves 

 differently from a double point. 



There is one respect in which the theory is deficient, and the 

 deficiency is the more important the larger the angular aperture. 

 The formula (7) from which we start assumes that a radiant point 

 radiates equally in all directions, or at least that the radiation from 

 it after leaving the object-glass is equally dense over the whole 

 area of the section. In the case of telescopes, and microscopes of 

 moderate angular aperture, this assumption can lead to no appreci- 

 able error ; but it may be otherwise when the angular aperture is 

 very large. The radiation from an ideal centre of transverse 

 vibrations is certainly not uniform in various directions, and 

 indeed vanishes in that of primary vibration. If we suppose such 

 an ideal source to be situated upon the axis of a wide-angled 

 object-glass, we might expect the diffraction pattern to be less 

 closely limited in that axial plane which includes the direction of 

 primary vibration than in that which is perpendicular to it. The 

 result for a double point illuminated by borrowed light would be 

 a better degree of separation when the primary vibrations are 



* The zero illumination extends to all points upon the line of symmetry. 

 Aug. 19th, 1903 2 h 



