with special reference to Corona and Iridescent Clouds. 275 



screen is so far away that our expression (2) still applies ; in 

 other words, the diffraction-pattern is not supplanted by a 

 geometrical image of the aperture. But the diffraction- 

 pattern is enormously reduced in size, and outside it there is 

 no light. In this outer region the illumination will be the 

 same whether we block up all the aperture except the original 

 slit, or block up the slit by an opaque filament leaving the 

 rest of the aperture open. For the two portions of light must 

 be able to neutralize each other. So we may replace the slit 

 by a filament of thickness a and length b, inclined to the inci- 

 dent light at the angle <y, lying within our large aperture, and 

 (2) will still hold good except within a negligibly small area. 

 If then, according to (3), at any point in the screen the light 

 is, say, green, to an eye placed at that point the aperture will 

 appear a green speck. As the green light is not in any way 

 altered by increasing the size of the aperture, it is clear that 

 it must come from the region immediately surrounding the 

 filament, and that the filament will look green even when the 

 diaphragm is entirely removed. 



Cloud of Filaments. 



The effect then is the same, whether it be produced by slits 

 in an opaque diaphragm or by filaments in an open space. 

 The calculations are simpler in the case of slits, but practically 

 we have to deal with filaments. So in future w r e shall speak 

 of filaments only, and in treating of the illumination of the 

 screen w r e shall refer only to the diffracted light and ignore 

 that which comes direct. 



As a is made small compared with b sin 7, the diffraction- 

 pattern is stretched out into a long strip, very narrow in the 

 y direction. If there be a large number n of filaments equally 

 inclined to the axis of z and evenly distributed round it, the 

 illumination is found by summing the illuminations due to the 

 individuals. Practically we have to distribute the light we 

 find, according to (2), on any circle round the axis of z evenly 

 over the whole circle and then multiply it by n. Owing to 

 the narrowness of the strip we can treat £ as constant for 

 points on the circle where the illumination is sensible. So of 

 the three factors on the right-hand side of (2), it is only the 

 last that varies. This third factor may be written p = sin 2 u/u 2 , 

 where u = irb smy.Tj/fX. If f and — f be the points where the 

 circle cuts the plane y = 0, the average value of p over the 

 whole circle is, remembering that the strip is cut twice, 



\pdn\TT%. 



The integration extends over the region for which p is sensible, 

 and we are of course at liberty to extend the limits to + oo . 



