from the Boundaries of Diffracting Apertures. 117 

 -expression (5) can be written in the form 



sm 



-J-COS27T 





Since the second aperture is also symmetrical about the 

 axis, the second integral is zero, for the elements of it 

 arising from two points situated at equal distances on 

 opposite sides of a diameter are equal and of opposite signs. 

 Therefore the intensity as viewed in the direction <£' is 



i= rrrv a vco S (2.^^.^ Jl (^./)i 2 . 



L Jo Jb i V A. / \ A/ /J 



Integrating with respect to </>', we get 



(neglecting a constant factor). 



If the angular semi-diameter of the lens be denoted by yjr, 

 then R=/^r. The expansion for the intensity can therefore 

 be written in the form 



c is a very large quantity, it is convenient to use 

 semi-convergent expansions for J and J x . We have 



r,. . /~2"r / 7r\f n 1 2 .3 2 1 2 .2 2 .5 2 .7 2 > 



Jo( - l) = V^L c03 (- i; -i)l 1 -2T(8¥) 2+ 4! (to)' — •} 



, . / i-\fl 1 2 .3 2 .5 2 1 , .3'.5*.7».9» 17 



+si y-viKv-juw + ^^ — ■••}]' 



©-••} 



3.5.7.9.1.3.5 



8.16.24.32 



, / 7r\(3 1 3.5.7.1.3/1V 

 + CQS (— 4JU-,- 8.16.24 U) 



,3.5.7.9.11.1.3.5. 



8.16.24.32.40 



ay--H 



