82 Lord Rayleigh on the Manufacture and 



the light proceeds be infinitely small, the image still consists of 

 a spot of finite size surrounded by dark and bright rings. That 

 this must be so may be shown by general considerations without 

 any calculations. If a lens is absolutely free from aberration, 

 the secondary waves issuing from the different parts of its hinder 

 surface agree perfectly in phase at the focal point. Let us con- 

 sider the illumination at a neighbouring point in the focal 

 plane. If the distance between the two points is so small that 

 the difference of the distances between the point under consi- 

 deration and the nearest and furthest parts of the object-glass is 

 but a small fraction of the wave-length (X), the group of secon- 

 dary waves are still sensibly in agreement, and therefore give a 

 resultant illumination the same as before. At a certain distance 

 from the focal point the secondary waves divide themselves into 

 two mutually destructive groups, corresponding to the nearer 

 and further parts of the object-glass. There is therefore here a 

 dark ring. Further out there is again light, then another dark 

 ring, and so on, the intensity of the bright rings, however, ra- 

 pidly diminishing. 



The radius r of the first dark ring subtends at the centre of 

 the Jens an angle 6 given by 



sin 0= -61^*, 

 K 



where R is the radius of the lens. If/ be the focal length, we 

 have 



-<■ 



Let us now suppose that the problem is to cover a square inch 

 with 3000 lines. On account of the curvature of the field it 

 would be impossible to obtain extreme definition over the surface 

 of a square inch with a less focal distance than (say) four inches. 



If we take /==4 and \= ■ _ # . ' ."we find 

 J 40,000 



K_ -61, 



10,000/ 



which gives 11 = % 2 for r— . That is to say, if the focal 



length were 4 inches and aperture '4 inch, the first dark ring- 

 corresponding to one of the lines would fall on the focal point 

 of the neighbouring one — a state of things apparently incon- 

 sistent with good definition. It is true that the aperture might 

 well be greater than half an inch, so that it may seem possible 

 to satisfy the requirements of the case. Rut the result of the 



* Verdet, Lemons d'Optique Physique, vol. i. p. 305. 



