216 Lord Rayleigh on Images formed 



disturbing definition was considered in my former paper. In 

 such a case as that last specified it is altogether negligible. 

 The advantage of a long focus was well understood by 

 Huyghens and his contemporaries; but it may have been 

 worth while to consider the matter for a moment from another 

 point of view, from which it clearly appears that the substi- 

 tution of an achromatic for a single lens serves no other pur- 

 pose than to diminish the minimum admissible focal length. 



Returning now to homogeneous light, let us consider the 

 case of an annular aperture of radii r x and r 2 . The extreme 

 difference of phase at distance /is now (?*— rf)-r-2f. If this 

 be £\, we get 



s _ 2(rj-rf) _ 2(r 2 + r 1 )(r 2 -r 1 ) 

 ■ h ~ V~ X ' 



as the value of the minimum distance at which a lens can be dis- 

 pensed with without loss. If r 2 — 1\ be small, f x is much smaller 

 than for a full circle of radius r 2 ; and it might appear that a 

 great advantage would be gained either in the diminution of 

 fx or by an increase in r 2 . The question, however, remains 

 whether with a lens the definition due to an annular aperture 

 of given outer radius r 2 is independent of the inner radius i\. 



The image of a mathematical point consists, it is known, of 

 a central patch of brightness, surrounded by rings alternately 

 dark and bright. If we conceive the radius of the central 

 stop (i. e. r x ) gradually to increase from to r 2 , the diameter 

 of the central luminous patch diminishes in the ratio 3*83 : 2*41. 

 From this it might be supposed that the definition due to the 

 marginal rim acting alone would be superior to the definition 

 due to the whole aperture*. It is true that there is at first 

 some improvement in definition ; but as r x approaches equality 

 with r 2 a rapid deterioration sets in, notwithstanding the 

 smallness of the central luminous patch. In order to under- 

 stand this it is necessary to examine more minutely the distri- 

 bution of light over the entire field. 



If the point under consideration be distant p from the centre 

 of the diffraction-pattern, the illumination for the full aperture 

 is given by 



2J i ( 27r vJ r_ «v 4 



2tt. 



p_ *r r 'v y/ y. *v 



Vfl „„!£ J"* 2 / 2 

 V 



* See a paper on the Diffraction of Object-glasses (Astr. Month. Notices, 

 1872). 



