88 Lord Rayleigh on Pin-hole Photography. 



the double radiant cannot be resolved in the image, unless the 

 angular interval exceed \/2r. 



Experiment* shows that the value thus roughly estimated 

 is very near the truth for a rectangular aperture of width 2?\ 

 If the aperture be of circular form, the resolving power is 

 somewhat less, in the ratio of about 1*1 : 1. 



It is therefore not going too far to say that there is nothing 

 better established in optics than the limit to resolving power 

 as proportional to aperture. On the other hand, the focal 

 length is a matter of indifference, if the object-glass be perfect. 



This is one side of the question before us. We now pass on 

 to another, in which the focal length becomes of paramount 

 importance. 



" The function of a lens in forming an image is to com- 

 pensate by its variable thickness the differences in phase 

 which would otherwise exist between secondary waves arri- 

 ving 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/ 1? 

 the extreme error of phase to be compensated falls to £X. 

 Now, as I have shown on a previous occasion f , an extreme 

 error of phase amounting to ^\, or less, produces no appre- 

 ciable deterioration 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 operation of increasing the focal length, the resolving 

 power of the instrument, which depends only upon the aper- 

 ture, remains unchanged; and we thus arrive at the rather 

 startling conclusion that a telescope of any 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, however, as we shall see, takes away almost all 

 practical importance from the proposition. 



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

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

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

 length must exceed, is given by 



V.'{/i , +^-/i=*X; 



so that [approximately] 



* " On the Resolving Power of Telescopes," Phil. Mag. August 1880. 

 t Phil. Mag. November 1879. 



