278 Mr. J. C. M c Connel on Diffraction- Colours, 



rings are given by 



z/tt=1-22, 2-23, 3*24, 4-24,... 

 and the maxima of the bright rings are given by 



e/7r=l-63, 2*68, 371, 4-72,... 

 and have the values 



0-0175, 0-00416, 0-00160, 0-00078, 

 that at the centre being unity. Thus corresponding parts 

 are rather further out than in the principal directions for a 

 square of side 2B, and the brightness falls off much more 

 rapidly. It seems legitimate to assume that the colours also, 

 when sunlight is used, are slightly purer. 



Cloud of Water-drops. 

 To pass from the illumination on a screen, due to a single 

 circular aperture, to the brightness of a water-cloud, we 

 follow the lines of the previous argument, with, however, 

 considerable simplification, owing to the orientation of a 

 sphere being a matter of indifference. We have to multiply 

 (9) by the number n of drops within an angular area equal to 

 that of the sun, and this number is given by 



mrI{ 2 = uf 2 co. 

 So the brightness of the cloud, in terms of that of the sun, i s 



«?vr -^M (10) 



The remarks we have made on (9) apply equally well to (10). 

 The colour-factor in both cases is Ji 2 (V). For the two kinds 

 of clouds, compare the values of the maxima given under (6) 

 and under (9). We are enabled to make a fairly complete 

 comparison by the following result. When z is great, 



Ji 2 0) = — sin 2 ^- j) nearly* 

 giving a colour-factor 



Xsin2 (*~£)- 



Even at the first bright ring the approximation is fair, for it 

 gives the first maximum at 0=l'7167r with the value 0'0162 ; 

 and it rapidly improves as z increases, though always better 

 at the maxima than at intermediate points. The expression 

 (6) may be written in the form 



a 2 sin 2 z x 

 * ' Wave Theory,' p. 432. 



