﻿the Micro- A.zl inometer. 397 



the illumination throughout a height / of the diffraction 

 pattern becomes sensibly independent of y, and in place of 

 (1) we have for the brightness 



B = B o sin 2 0/^ 

 where 6 = irax/\f J* * 



27. Since in practice the width of the slit must be finite, 

 let X be a coordinate measured horizontally in the plane of 

 the slit, and let 



cfi^iraX/Xf; (3) 



the width of the slit extending from $——$ to ^ = /3. 

 Corresponding to any elementary width d<p of the slit, the 

 brightness at any position 6 in the focal plane is 



C^># ; (4) 



where is a constant which contains as a factor the bright- 

 ness of illumination of the slit*. At any point whose 

 coordinate is #, the resultant brightness is found by inte- 

 grating (4) from </>=— {3 to </>==/3; and as we shall only be 

 concerned with values of 6 fairly close to +7r, while tor a 

 slit suitably narrowed for observation $> is small, no very 

 great error will be introduced by taking the denominator 

 (0 — (f>) 2 as constant throughout the integration and equal to 

 7r 2 . Putting, for example, 



6= — 7r 4-7 where 7 is small, 



the integral has the approximate value 



^(2 7 */3 + §/3 3 ), 



which, to our degree of approximation, is a minimum when 

 7 = 0; i. <?., when Q— — tt. 



* Here, and in -what follows, it is supposed that light is focussed on 

 the slit under a sufficiently wide angle to ensure the full theoretical 

 brightness, notwithstanding the spreading of the transmitted pencil by 

 diffraction. In practice it -would not be convenient to narrow the slit 

 beyond a certain limit; when a very bright source was available, advan- 

 tage would rather be gained by increasing the focal length f\ a cylin- 

 drical lens being used as eyepiece. It must, however, be remarked that, 

 in these circumstances, the apparent vertical height of the pattern is 

 inversely proportional to f(cf, § 22), 



