276 Mr. J. 0. M c Connel on Diffraction- Colours, 



C + ® gin 2 u 

 Now 1 — j- du =7r is a known result, and 



%J — CO 



rrb sin y drj/f\ = du. 



"° J pdr}/7r^ =/X/7rb sin 7 f . 



Hence the illumination on the screen at points in the plane 

 3/ = 0, due to a large number n of equal regularly distributed 

 filaments making an angle y with the axis of z, is by (2), 



2 . .sir -2i— 



^6siii7 / /\ . . 



7T/ 2 X f TrVf W 



To extend this to the case when the filaments are uniformly 

 distributed in all directions we must replace nb sin 7 by S&sin 7. 

 If they occupy the fraction a of the field of view looking from 

 the screen, and the summation be extended over an angular 

 area o> equal to that of the sun, we have 



acof 2 = %ab sin 7 = aZb sin y . 

 Now the direct illumination of the sun at the cloud is, by 

 hypothesis, unity, and it has the same value where the observer 

 stands, i. e. at the imaginary screen. And it is obvious that 

 the apparent brightness of the sun and cloud are in the same 

 ratio as the illumination due to equal angular areas of each. 

 So, finally, the brightness of the cloud of filaments is in terms 

 of that of the sun, 



af f\ 



TZB ' .".... (6) 



On p. 431 of my former article are given expressions for 

 the brightness of the first, second, and fourth bright rings in 

 a cloud of filaments, obtained in a different manner. It will 

 be found on examination that these expressions agree with (6). 



In a cloud of filaments, of diameter a, the first four maxima, 

 according to (6), are proportional to 1, 0*215, 0*076, 0'035 ; 

 the ninth being 0*0036, and the central maximum being infi- 

 nite. This last result is not surprising, for we have supposed, 

 throughout the greater part of the argument, the source of 

 light to be indefinitely small. 



In sunlight the colour is defined by the factor 



Xsm ji ( 7 > 



These are the colours produced when the source is a luminous 



