THE KAINBOW DUE TO A CIRCULAR SOURCE OF LIGHT. 19 



Airy's theory applies ns well as in the case of the spherical 

 drop, when we change the vahie of A 



/(^^)=const. \'f(xd) 



where ^-(^-)'- 



If we take into account the l)readrh of the slit, there is no 



difficulty in applying reasoning similar to the above, to arrive 



at the expression 



1 





^Yhere 2^1'= the angular breadth of the slit as viewed from the 

 centre of the glass rod. Or putting z=y.(f , 



- <i> 



This coincides with the integral at [a) on which Pernter's cal- 

 culation was based. Hence, we see that Pernter's integral holds 

 good for the case of slit and cylinder, but not for the case of 

 circle and sphere. 



The difference of the expression of A in the two cases of 

 sphere and cylinder was not discussed by Airy and others ; but 

 the existence of the difference is evident from the geometrical 

 theory of the rainbow, in which the intensity is proportional to 

 r" in the case of the sphere and to r in the case of the cylinder. 



In § G we always substituted the mean value of V {-/.ipf—z- 

 before integration, so that the expression foi- F lioeonies only 

 roughly approximate; but in the present case, ihere being no 



