by Apertures with Curvilinear Boundaries. 297 



Therefore 



c t' f* °° /' C °° ^ 



S = /' dy \ sin 2ir ^ I cos - } v 2 dv + cos - 7r m I sin ^ ?> 2 tZv [ , 



r here 



/' = 



A' 



d v ' = \/i{i + l)*' cos< i>- 



By using the semi-convergent expansions, we have 



(TOO 

 oos^»*dv - Mcos^' 2 -Nsin^' 2 , 



I sin ~ » 2 c^ = M sin | */ 2 + N cos ~ w' s 



where 



'2. 



M 1 _1.3.5 JL 3.5.7.9 



, xr X 1.31.3.5.7 

 and N = — ; ^ + r-nr-. 



7TV TT°V 



77" V 



As we are dealing with large-angle diffraction, v is 

 large and it is sufficient to retain the first term only, and 

 we may write 



f * 2 7 1 ^* /2 1 • 7T ,o 



1 cos -r raw = ~TT3 cos 7* v , sin — v 2 , 



f- 7r 27 1 ' ^ /2 . 1 ^ /9 



1 sm -x v z dv — 9 ,„ sin — v " H / cos 77 v' 2 . 



J v > 2 7rV 3 2 7TV 2 



Consequently, the resultant amplitude at P 



Neglecting I -xTa)' which is very small compared with 

 ( — jA , the amplitude is equal to 



f'dy _ A Kdy 



ttv' ~ 2ir(pi -hpoW cos 2 (f)' 



