172 Lord Bayleigh on the Theory of Optical Images, 



elements of the grating to possess permanent phase-relations. 

 It appears, therefore, to be a desideratum that the matter 

 .should be reconsidered from the older point of view, according 

 to which the typical object is a point and not a grating. Such 

 artreatment illustrates the important principle that the theory 

 of resolving-power is essentially the same for all instruments. 

 The peculiarities of the microscope arise from the fact that 

 the divergence-angles are not limited to be small, and from 

 the different character of the illumination usually employed ; 

 but, theoretically considered, these are differences of detail. 

 The investigation can, without much difficulty, be extended 

 to gratings, and the results so obtained confirm for the most 

 part the conclusions of the spectrum theory. 



It will be convenient to commence our discussion by a 

 simple investigation of the resolving-power of an optical 

 instrument for a self-luminous double point, such as will be 

 applicable equally to the telescope and to the microscope. In 

 fig 2 AB represents the axis, A being a point of the object 

 and B a point of the image. By the operation of the object- 

 glass LL/ all the rays issuing from A arrive in the same phase 

 at B. Thus if A be self-luminous, the illumination is a 

 maximum at B, where all the secondary waves agree in phase. 



Fisr. 2. 



F3- 



B is in fact the centre of the diffraction disk which constitutes 

 the image of A. At neighbouring points the illumination is 

 less, in consequence of the discrepancies of phase which there 

 enter. In like manner, if we take a neighbouring point P in 

 the plane of the object, the waves which issue from it will 

 arrive at B with phases no longer absolutely accordant, and 

 the discrepancy of phase will increase as the interval AP 

 increases. When the interval is very small, the discrepancy 

 of phase, though mathematically existent, produces no prac- 

 tical effect, and the illumination at B due to P is as important 

 as that due to A, the intensities of the two luminous centres 

 beino- supposed equal. Under these conditions it is clear 

 that A and P are not separated in the image. The question 

 is to what amount must the distance AP be increased in 

 order that the difference of situation may make itself felt in 

 the image. This is necessarily a question of degree ; but it 

 does not require detailed calculations in order to show that 



