258 On the Visibility of Interference-Fringes. 



Putting fca ll fi = §, «a 1] /? , = A: ll7 and expanding, 



2I=$I*dady 4 cos $ jTl 2 cos k n xdxdy + sin 3 jj I 2 sin k n xdxdy . (5) 



Let y = (/}(ai) be the equation of the curve bounding the 

 luminous surface; or, better, let c£>{x)dx be the "total 

 intensity " of a strip of width dx. 



Denoting I I 2 dy by F( t i'), and omitting the factor 2, 



equation (5) becomes 



T=\¥(x)dx-\- cos S f F(#) cos kxdx+ sin $ f F(#) sin kxdx, 



or I = P + Ccos3 + Ssin3 (6) 



If the width of the apertures is small compared with their 

 distance, the variations of F(#) with //< (or $) may be neglected, 

 and in this case the maxima or minima occur when 



s 



tan $ = ^, or when I = P + y' C 2 + S 2 - 



If now the visibility of the interference-fringes be defined 

 as the ratio of the difference between a maximum and an 

 adjacent minimum to their sum 



or 



[ J F (a?) cos kx dx] 2 + [ j" F(x) sin kx dx] 2 



[$F(x)dx]* ~* ' ' (?) 



For narrow rectangular apertures, 



In this expression, if v=0 and b = length of aperture, 



^J = 9I fa W aild ^2 = S- &(#). 



So long as 



2^) X 

 r < 6' 



F(a?) is nearly proportional to <j>(ai) ; that is, so long as the 

 angle subtended by the source is less than the limit of resolu- 

 tion of a telescope with aperture b, the brightness is pro- 



