THE RAINBOW DUE TO A CIRCULAR SOURCE OF LIGHT. 11 



and F, the difterence between a point and a circular source is to 

 be found. f'{-<d) does not change for different values of /, if we 

 reduce the scale of 6 properly, because it is a function of xß, 

 but not a function of /. and 6 taken separately ; this however, 

 does not hold for F, which is a function of /. as well as of xO. 

 Thus it is necessary to consider F more in detail, though its 

 evaluation as a function of x, O, and xO is by no means easy. 

 If we try to expand f-<x{d + ipQ,os,il>)l in a power series of 

 x(fao^(l', then its coefficients gradually increase with xO, and are 

 very inconvenient for values oî /.tJ>\. If we change f-\x{i^ + ipiioi^d>)i 

 to a double integral 



= -_^ I dx i dy\cos-^<x^+)/—xd{x + y)icos-^\x(pcos(/>{x + y)> 



— sm-^}x"'+y-'~-xd(x + y)>sïn-^\x<pcosô{x+i/)^ 

 + cos- '- <x^ — ?/" — xd(x — y) \ cos-^< /.(p cos (p{x —y)> 



- sin-^|j3^~r-^'^(-«-2/)|si"-9-{^^cos^G^'--^)j] > 



and integrate with respect to (f and </•, by using the relations of 

 Bessel's function 



J,^(?y)= I C0S{W cos f2)d /J. , 



0= I sin(?rco?/^)r/// , 

 and -j^^. U-^Jiiv ^<^)^= .j-^oiiAî^) > 



