458 Mr. L. K. Gr. Filon on certain Diffraction Fi 



if we only consider the appearances along the line /> = 0, taken 

 to pass through the centre of the image, which is assumed 

 circular or elliptic. 



Denoting -yr— by jjl and expanding, we get 



sm *^ - r-1 2(r-l) 



frX x 2W , , (-1) (2H 



/2ttA^\ 2 ~ 2 ' 3! + ' * * + *-.(2r-l)I " t "* ,,? 



V bX 



whence 



32Awf (7 1 (2,0V, f-lHy 1 *" , ) 



4™(£+t0 rf 1 



X cos 



OX 



where 



taken all over the source, and is essentially positive and in- 

 dependent of a and q. 

 Now 



| w sin — — — tf# = 



for a circle or ellipse. 



To find the cosine integral, C, we have, d and ot having 

 the same meaning as before, 



C = 4 — 2r -i v& 1 — o 2 ) cos —r—dv. 

 Jo w 6\ 



Put v = «r cos 0. 



C = 4^-^J sin* d cos (^cos tf)<tf 



T (±ira\ 



/47T05Y ^ ^ ' 



T /47TC/'5r\ 



_ 2fl? a - 1 i8nr(2r-l) ! A ^ j 



2 r ~ 1 (r — 1)! /47raOT\ r 



