416 Transactions of the Society. 



tinguishable, as you like. Helmholtz thought that this point 

 would be reached when the two antipoints overlap to the extent 

 of one semi-diameter, i.e. when the edge of the one lies exactly on 

 the centre of the other. 



For this conclusion Helmholtz gives no physical reason, and no 

 physical reason can be found. On the contrary, it can be shown — 

 and I have developed the argument in the appendix (Note II.) to 

 this paper — that there is strong ground in physics for concluding 

 that antipoints may overlap to a much greater extent than that of 

 the semi-diameter, and still yield fully resolved pictures. But 

 Helmholtz seems to have relied upon experiments, and I shall best 

 do justice to his paper if I quote it textually in this connection.* 



" We are here mainly concerned with the diffraction images 

 which arise from an aperture of circular form. A bright point of 

 light (the reflection of the sun from a thermometer bulb) seen 

 through such an opening (a needle- prick in a card) appears, as is 

 known, like a bright circular disc surrounded by rings alternately 

 dark and bright. The apparent breadth of these rings reckoned 

 from minimum to minimum, corresponds very closely to a visual 



angle of which the sine is -,, where X is the wave-length of the 



incident light, and d the diameter of the opening. The outermost 

 rings have almost exactly this breadth, the innermost are slightly 



broader ; the radius of the central light disc is 1 ■ 220 - . Assuming 



Cv 



that the smallest visual angle under which we can distinguish two- 

 fine bright lines from one another may be set down as one minute 

 of angle, then fringes of the brightest greenish-yellow light, having 

 a wave-length of 0*00055 mm., are visible when the diameter d of 

 the opening is 1'89 mm. The spreading out of a bright point into 

 a disc or a bright line into a stripe must evidently become notice- 

 able with somewhat wider openings. 



" If objects having distributed bright patches of surface are 

 viewed through such openings, the diffraction figures of the several 

 points of light upon such a surface will tend to overlap one 

 another partially, so that the circular fringes of every several point 

 cannot be separately recognised. Now it is clear that this result 

 of diffraction which changes every point of light into a minute 

 circular disc, must impair the definition of the object as definition 

 is impaired in the eye by the minute diffusion circles which result 

 from imperfect accommodation. Very small objects which are 

 only discernible in the most sharply defined retinal images will 

 then become unrecognisable. 



" That this is so can be proved by a simple experiment. The most 

 critical objects are gratings having alternate bright and dark stripes. 



* Togg., p. 570 ; Abh., p. 198. 



