41: R Notes. 



be removed from its seat, so that A is no longer focused upon the 

 object-glass, the definition must recover. 



I do not know how far the above reasoning will seem plausible 

 to the reader, but I may confess that I was at first puzzled by it. 

 I doubt whether any experimenter would willingly accept the 

 suggested conclusion, though he might be unable to point out a 

 weak place in the argument. He would probably wish to try the 

 experiment ; and this is easily done. The lens L may be the 

 collimating-lens of an ordinary spectroscope whose slit is backed 

 by a flame. The telescope is removed from its usual place to a 

 distance of say 10 feet and is focused upon L. The slit is at the 

 same time focused upon the object-glass of the telescope. 

 Although the image of the slit is very narrow, the definition of L 

 as seen in the telescope does not appear to suffer, the vertical parts 

 of the circular edge (parallel to the slit) being as well defined as 

 the horizontal parts. If, however, at the object-glass a material 

 screen be interposed provided with a slit through which the image 

 of the first slit can pass, the definition at the expected places 

 falls off greatly, even although a considerable margin be allowed 

 in the width of the second slit. 



This experiment gives the clue to the weak place in the 

 theoretical argument. It is true that the greater part of the light 

 ultimately reaching the eye passes through a very small area of 

 the object-glass ; but it does not follow that the remainder may be 

 blocked out without prejudice to the definition of the boundary of 

 the field. In fact, a closer theoretical discussion of the diffraction 

 phenomena leads to conclusions in harmony with experiment. 



In the case of a point-source and the complete circular aperture 

 LL, the question turns upon the integral 



/. 



J (a x) J\ (/3 a?) d x, 



J , Ji being the Bessel's functions usually so denoted. The 

 integral passes from to 1//3, as a passes through the value /?*. 



If the aperture of LL be reduced to a narrow annulus, the 

 integral to be considered is 



L 



J (a x) J (/3 x) xdx. 



This assumes an infinite value when a = /3 f- 



If the apertures be rectangular, the integrals take still simpler 

 forms. 



* A theorem attributed to Weber See Gray and Matthews' " Bessel's Func- 

 tions," p. 228. 



t See " Theory of Sound," § 203, equations (14). (16). 



