182 Lord Rayleigh on the Theory of Optical Images, 



geometrical images. The only difference in the calculations 

 is that represented by the substitution of 2J X for sine. We 

 shall not, however, occupy space by tables and drawings such 

 as have been given for 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= — 7r, these being the points just mentioned and 

 that midway between them at u=— \ir. The values of the 

 functions required are 



2J 1 (0)/0 =1-0000= ^ {1-0000}. 



2J 1 {7r)/7r = -1812= v/{-03283}. 



2Ji(i7r)/i7r = -7217= ^{-5209}. 



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

 larity of phase, we have at the geometrical images amplitudes 

 1*1812 as against 1*4431 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 >/{ 1*0328} or 

 1*0163, and at the middle point v '{ 1*0418} or 10207. The 

 partial separation, indicated by the central depression in 

 curve iii. fig. 4, is thus lost when the rectangular aperture is 

 exchanged for a circular one of equal w r idth. It should be 

 borne in mind that these results do not apply to a double line, 

 which in the case of a circular apertui'6 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 appreciable 

 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 dif- 

 fraction pattern to be less closely limited in that axial plane 



* The zero illumination extends to all points upon the line of sym- 

 metry. 



