441: Mr. L. N. G. Filon on certain Diffraction Fringes 



Hence the difference of retardation measured by length in 

 air 



P'Q-SQ=const. +(| + ^)» + (f + \)y 



+*(*+/>(? -If)- 



If now the geometrical image of S lie on the screen AP 

 {i.e. if the screen is in correct focus) b' = W and the last 

 term disappears. 



If, however, the screen be out of focus \jV is not equal to 

 1/W, and the term in x^+y 2 may be comparable with the 

 two others, if b' be not very great compared with b. Thus 

 we see that appearances out of focus will introduce expres- 

 sions of the same kind as those which occur when no lenses 

 are used. 



We will, however, only consider the case where the screen 

 is in focus. Let — u and — v be the coordinates of the geome- 

 trical image of S ; then 



u/b = TJ/W, v/b=Y/W. 



The difference of retardation measured by length in air is 

 therefore of the form 



Hence the total disturbance at P (integrating over the two 

 slits) is given by the expression 



a+k h 



a -k —h 

 -a + k h 



C j Cj ' 27r / xt . P + n . 9 + v \ 



A 



■a — k 



where A = a constant, 



2a = distance between centres of slits, 

 2 k = breadth of either slit, 

 21i — length of either slit. 



This, being integrated out, gives 



^ 2ALX . 2irt 27rq+v . 2irq+v, . Zirp + u, 



D=— 57 — i — \7 — r~\ sm — cos— -^-^— asm— ^— Asm r ~* h, 

 ir 2 (p-\-u)(q + v) r X b X b X b 



