THE RAINBOW DTE TO A CIRCULAR SOURCE OF LIGHT. 5 



when we confine our attention to small values of ^ ; and then 

 the intensity of light in the direction o is given by 



i{d) = const. A/-(/ß) (3) 



where A = {-^f , (3„) * 



/{xd) = I cos --^(ii^ — xdu)dic. (3g) 



But, if we do not neglect the visual angle of the drop, the 

 definition of Ö must be slightly changed. In this case the ray 

 of minimum deviation does not pass c, but meets the surface of 

 the drop at a point say c. Thus c' must be taken as the 

 coordinate origin and co the direction of the observer ; then 6 



is defined by 



6 = |o'c'm. (!') 



Using this value of ß, and neglecting rß compared with the 

 observer's distance, we may state the same formula as the above. 

 For different wave lengths of light, the point c' is slightly dis- 

 placed, but the amount of the displacement being negligibly 

 small, we may take one })Osition of c' as the coordinate origin 

 for all the wave lengths of a visible ray. 



Airy expanded f{xß) as a power series of x6, which is not 

 convenient for a practical calculation of values for ^ß>o, though 

 it always remains convergent. On the other hand, especially 

 for large values of 6, the following semiconvergent series taken 

 from fStokes''^ can be employed with advantage : — 



/{y.6)=-2h-^ixd)-* M CU8 (^;,__^-^j , 



u J ^^ ¥ 



where ^~ l~T"j ' 



(1) Collected Papers, 11. p. :J2U (Lundon, iSbo). 



