Cause of Iridescence in Clouds. 431 



sun is cfib 2 . A/X 2 r 2 by Stokes's formula*, and the intensity of 

 the first spectrum is then, according to Yerdetf, 



•046a 2 6 2 .A/W. 



While, if the slit or filament be inclined to the sunlight at an 

 angle j3, the intensity is 



•046 a 2 6 2 sin 2 /3 . A/W. 



By what we have just shown, we need only consider a frac- 

 tion, viz. X/46siny, of the filaments to send light, and, since 

 the effectiveness of each is proportional to sin 2 /3, we must 

 diminish the final product in the ratio 2 : 1. (The correct 

 ratio is 2 : l + sin 2 7, but in our applications 7 is small.) 

 Hence the average filament sends fight of intensity 



A .„a 2 6 2 . A, 

 •046-t-tt.A 



\V 86 sin 7* 



Now we want to compare the brightness of the cloud with 

 that of the sun. Let co be the angular magnitude of the sun, 

 and let the filaments occupy a fraction a of the field of view. 

 The average apparent size of a filament is irahj^. So, if n be 

 the number of filaments in the area 6>, we have 



cnr' 2 co=n7rab/4:. 



Hence the intensity of the fight from the n filaments is 



*023 acou . A/Xtt sin 7, 



and the brightness of the cloud, in terms of that of the sun, 

 is 



*023 awci/Xir sin 7. 



Now for the first spectrum, 



a/X=l*43/sin^, 



where -\|r is the angular distance from the sun, and so -^r = 2y. 

 So, inserting numerical values, the last expression takes the 

 form 



•0041 a/^ 2 , 



where ^ is supposed to be less than 30°, and is expressed in 

 degrees. 



In the same way we find for the second and fourth spectra 

 the expressions 



•0026 «/^ 2 



•0015 cc/yjr^. 



* Stokes, " On the Dynamical Theory of Diffraction," Trans. Canib. 

 Phil. Soc. yoI. ix. p. 1 ; or 'Math, and Phys. Papers,' vol. ii. p. 243; or 

 Glazebrook, " On Optical Theories," B. A. Report, 1885. 



t Loc. cit. 



