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XXIX. On Images formed without Reflection or Refraction. 

 By Lord Rayleigh, F.R.S., Professor of Experimental 

 Physics in the University of Cambridge* . 



THE function of a lens in forming an image is to compen- 

 sate by its variable thickness the differences in phase 

 which would otherwise exist between secondary waves arriving 

 at the focal point from various parts of the aperture. If we 

 suppose the diameter of the lens (2r) to be given, and its focal 

 length / gradually to increase, these differences of phase at 

 the image of an infinitely distant luminous point diminish 

 without limit. When/ attains a certain value, say f l9 the 

 extreme error of phase to be compensated falls to |X. Now, 

 as I have shown on a previous occasionf, an extreme error of 

 phase amounting to ^A,, or less, produces no appreciable dete- 

 rioration in the definition; so that from this point onwards the 

 lens is useless, as only improving an image already sensibly 

 as perfect as the aperture admits of. Throughout the opera- 

 tion of increasing the focal length, the resolving-power of the 

 instrument, which depends only upon the aperture, remains 

 unchanged; and we thus arrive at the rather startling conclu- 

 sion that a telescope of an}^ degree of resolving-power might 

 be constructed without an object-glass, if only there were no 

 limit to the admissible focal length. This last proviso, how- 

 ever, as we shall see, takes away almost all practical import- 

 ance from the proposition. 



To get an idea of the magnitudes of the quantities involved, 

 let us take the case of an aperture of 5- inch, about that of the 

 pupil of the eye. The distance f 1} which the actual focal 

 length must exceed, is given by 



so 



that 





2 ^ -/1 



= i\; 



Tl 



ms, if X,= 



1 ™_ _1_ 



40000? ' — 10? 



/i= 



800. 



The image of the sun thrown on a screen at a distance exceed- 

 ing 66 feet, through a hole ^ inch in diameter, is therefore at 

 least as well defined as that seen direct. In practice it would 

 be better defined, as the direct image is far from perfect. If 

 the image on the screen be regarded from a distance f lf it will 

 appear of its natural angular magnitude. Seen from a dis- 



* Communicated by the Author, 

 t Phil. Mag. November 1879. 



