The Theory of Optical Images. By Lord Raylcigh. 451 



valuable as it is, needs a good deal of supplementing, even when 

 the representation of a grating under parallel light is in question. 



When the object under examination is not a grating or a 

 structure in which the pattern is repeated an indefinite number of 

 times, but for example a double point, and when the incident light 

 is not parallel, the spectrum theory, as hitherto developed, is 

 inapplicable. As an extreme example of the latter case we may 

 imagine the grating to be self-illuminous. It is obvious that the 

 problem thus presented must be within the scope of any complete 

 theory, and equally so that here there are no spectra formed, as 

 these require the radiations from the different elements of the 

 grating to possess permanent phase-relations. It appears, there- 

 fore, 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 a treatment illustrates the 

 important principle that the theory of resolving-power is essentially 

 the same for all instruments. The peculiarities of the microscopt 

 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 differ- 

 ences 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. 118 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. 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 practical effect, and the 

 illumination at B due to P is as important as that due to A, the 



the view taken, any theory of resolving power of optical instruments must be a 

 diffraction theory in a certain sense, so that the name is not distinctive. Diffraction 

 is more naturally regarded as the obstacle to fine definition, and not, as with some 

 exponents of Prof. Abbe's theory, the machinery by which good definition is brought 

 about. 



