THE RAINBOW DUE TO A CIRCULAR SOURCE OF LIGHT. 15 



Then, the line T^ does not pass the centre of the source, but 

 is approximately parallel to Do. Representing the angular dis- 

 tance of L from Dq by y, and the angular length measured 

 from the middle point of Ij along L by x, we have as the in- 

 tensity due to L in the direction 6 in the plane Leo, 



I(ö)=--const. A f f-\x{d-Vx)\,lx . {b) 



This value of I{d) being a function of y, holds for any plane 

 which contains a part of the source and c, o. For a particular 

 value y=o, (b) reduces to (a), and when y = ^, I(^) becomes 

 zero. 



The above discussion shows that, in general, when both the 

 angular diameters of the source and of the drop arc not negli- 

 gible, the intensity of the rainbow can not be considered as 

 uniform along the arc of the rainbow ; and exact investigation 

 is almost impossible unless we are informed of the distribution 

 of the drops. Again, if we consider the drop so large that the 

 distance of the lines AiO, AoO in Fig. (2) is greater than the 

 pupil of the observer's eye, the result must be considerably 

 changed. 



If we take the mean value of intensity I(^^) along the arc 

 of the rainbow which is specified by 6 as 



1/0)2- 



y' 



IJd) = const. A f J Mx{d + x)\ dy dx , 



" -i/<I)- — ?/- 



changing the order of integration, we have 



IJi9)=const. A r dxJ<^''-x'Ax{d + x)\ 



