without Reflection or Refraction. 217 



if y = 27r^, Ji being the symbol of the BesseFs function of 



A/ 

 order unity. The dark rings correspond to the roots of J l5 

 and occur when y = 3"83, 7*02, 10*17, &c. 



The whole illumination within the area of the circle of 

 radius p is given by 



fl 2 27r^ = 27rr 2 (V'JJCy)^. 



This integral may be transformed by known properties of Bes- 

 sel's functions. Thus*, 



y dij 



so that 



^W oW .j lW _j l( ,)^M 



We therefore obtain 



KyXy=l-J&/)-J?(2/)- 



Jo 



If y be infinite, Jq(?/) and Ji(y) vanish, and the whole illu- 

 mination is expressed by irr 2 , as is evident a priori. In general 

 the proportion of the whole illumination to be found outside 

 the circle of radius p is given by 



J%0 + J?(y)- 



For the dark rings J 1 (y) = 0; so that the fraction of illu- 

 mination outside any dark ring is simply Jj(y). Thus, for the 

 1st, 2nd, 3rd, and 4th dark rings we get respectively *161, 

 •090, '062, and -047, showing that more than -^ of the whole 

 light is concentrated within the area of the second dark ring. 

 The corresponding results for a narrow annular aperture 

 would be very different, as we may easily convince ourselves. 

 The illumination at any point of the central spot or of any of 

 the bright rings is proportional to the square of the width of 

 the annul us, while the whole quantity of light is proportional 

 to the width itself. As, therefore, the annulus narrows, a less 

 and less proportion of the whole light is contained in any finite 

 number of luminous rings, and the definition of an image cor- 



* Todhunter's Laplace's Functions, p. 297. 



