174 Lord Rayleigh on the Theory of Optical Images, 



linear measure of aperture and an angular limit of resolution, 

 whereas in the case of the microscope the limit of resolution is 

 linear and it is expressed in terms of angular aperture. 



In the above discussion it has been supposed for the sake of 

 simplicity that the points to be discriminated are self-lumi- 

 nous, or at least behave as if they were such. It is of interest 

 to inquire how far this condition can be satisfied when the 

 object is seen by borrowed light. We may imagine that the 

 object takes the form of an opaque screen, perforated at two 

 points, and illuminated by distant sources situated behind. 



If the source of light be reduced to a poiut, so that a single 

 train of plane waves falls upon the screen, there is a perma- 

 nent phase-relation between the waves incident at the two 

 points, and therefore also between the waves scattered from 

 them. In this case the two points are as far as possible from 

 behaving as if they were self-luminous. If the incidence be 

 perpendicular, the secondary waves issue in the same phase ; 

 but in the case of obliquity there is a permanent phase- 

 difference. This difference, measured in wave lengths, in- 

 creases up to e, the distance between the points, the limit 

 being attained as the incidence becomes grazing. 



When the light originates in distant independent sources, 

 not limited to a point, there is no longer an absolutely definite 

 phase-relationship between the secondary radiations from the 

 two apertures ; but this condition of things may be practically 

 maintained, if the angular magnitude of the source be not too 

 large. For example, if the source be limited to an angle 

 round the normal to the screen, the maximum phase-difference 

 measured in wave-lengths is esin 0, so that if sin 6 be a small 

 fraction of X/e, the finiteness of 6 has but little effect. When, 

 however, sin 6 is so great that e sin 6 becomes a considerable 

 multiple of X, the secondary radiations become approximately 

 independent, and the apertures behave like self-luminous 

 points. It is evident that even with a complete hemi- 

 spherical illumination this condition can scarcely be attained 

 when e is less than X. 



The use of a condenser allows the widely-extended source 

 to be dispensed with. By this means an image of a distant 

 source composed of independently radiating parts, such as a 

 lamp-flame, may be thrown upon the object, and it might at 

 first sight be supposed that the problem under consideration 

 was thus completely solved in all cases, inasmuch as the two 

 apertures correspond to different parts of the flame. But we 

 have to remember here and everywhere that optical images 

 are not perfect, and that to a point of the flame corresponds 



