450 Mr. L. N. G. Filon on certain Diffraction Fringes 



which, on being integrated, gives 



2kbX . 2-rrq + Vj . 2-rrp + u, . 2ir/Xt \ 2iry 



"27 — : \ f , — >sm -r- - T ™ A: sm — ^— h sin — ( e Icos , 



7r 2 (p-j-u)(q+v) X b \ b \\t J X 



where — €-y=— V -Jl a -2ftcosf, _ 6 + 7= ?; -±-£ a _2«cos<£, 



whence e = a cos <£ -f ft cos >/r, 



7= — t— a + p cos >/r — a cos </>. 



Hence the intensity I of light at (p, q) is 



ir (p + u)*(q + vr X b X b X 



, v + 2 Scos 6' — a cos z (ft sin 0' -\- a sin 0) 



where 7= -^-^a + ~ / o b— x a 



b x /f + q 2 + b 2 </ p 2 + q * + b * 9- 



In the last term we may put — y =^ = |-, for if we went 



Vi? 2 + g 2 + b 2 o 



to a higher approximation, we should introduce cubes of p/b, 

 qlb which we have hitherto neglected. 



If, further, we make ft cos 6' = a cos 0, which can always he 

 managed without difficulty, the second term, which would 

 contain squares on expansion, disappears and we have 



(v+ q)a — ( ft sin 0' + asin 6)q 

 7 _ 



= \va + q(a-(ft sin 0' + a sin 6))\/b. 



This gives fringes of breadth bX/2(a— (ft sin & + a sin 0)). 

 These may be reckoned from the bright fringe 7 = ; i. e. 



— va 



qQ = a-{ftsm6' + *sm0)' 



The visibility of the fringes for a single source will, as before, 

 not be affected by changing a : for a second source the origin 

 of the fringes is given by 



q '=—v'a/(a—^ft sin 0' + asin 0^). 



and if the visible rectangles overlap, there will be a sensible 

 diminution of the fringe appearance whenever 



q - q Q ' = (n + i)bX/2\a- (ft sin 0' + a sin 0) } 



where n is an integer; 



i.e. v' — v = an odd multiple of bX/4a, 



the condition previously found. 



