The Theory of Optical Images. Bij Lord Bayleigh. 453 



the object-glass, and D the distance of the object, the angle 

 subtended by AP is e/D, and the angular resolving-power is given by 



2Dsina"2E' K) 



the well-known formula. 



This method of derivation makes it obvious that there is no 

 essential difference of principle between the two cases, although 

 the results are conveniently stated in different forms. In the case 

 of the telescope we have to do with a linear measure of aperture 

 and an angular limit of resolution, whereas in the case of the 

 microscope the limit of resolution is linear and 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-luminous, 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 point, so that a single 

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

 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 second- 

 ary waves issue in the same phase ; but in the case of obliquity 

 there is a permanent phase-difference. This difference, measured 

 in wave-lengths, increases 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 aper- 

 tures ; but thisjeondition 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 6 round the normal to the 

 screen, the maximum phase-difference measured in wave-lengths is 

 e sin 6, so that if sin 6 be a small fraction of \Je, the finiteness of 

 6 has but little effect. When, however, sin 6 is so great that e sin 6 

 becomes a considerable multiple of A., the secondary radiations 

 become approximately independent, and the apertures behave like 

 self-luminous points. It is evident that even with a complete 

 hemispherical illumination this condition can scarcely be attained 

 when e is less than A. 



The use of a condenser allows the widely-extended source to be 



