Rainbow due to a Circular Source of Light. 601 
where 
F(K, ®, K@) = aol" i f bdddrbf2{K (0+ bcos) }, 
F*{K(O+¢ os et cos - fy—K(0 + ¢ cos w)updu. 
Thus the function f? in Airy’s theory is replaced by a more 
general function F. From the form of the functions /? 
and HF’, the differences between a point and a circular source 
are to be found. f?(K@) does not change for different values 
of K, if we reduce the scale of @ properlv, because it isa 
function of K9, but not a function of K and @ separately; this, 
however, does not hold for F, which is a function of K as well 
as of Ké. Thus it is necessary to consider F more in detail, 
though its evaluation as a function of K, ®, and K@ is by no 
means easy. 
If we try to expand f7{K(@+ cos y)} in a power series 
of K¢ cos w, then its eocticients gradually increase with K@, 
and are very inconvenient for values of K9>1. If we change 
F7{K(0+ cos y)} to a double integral 
a) oO 
| cos 5 (a®—K(0+ ¢ cos ye}ae |” cos 5 {y° —K(6+¢ cosp)ytdy 
gi aay tae (see Nee 4 . 
=s) ae | ay | cost {a8 +y— Kees y)} cos F {Ko cos y(w+y)} 
0 0 s 
analy +y°—Kd(«+y) ) } sin 5 (Ko cos (e-+y) } 
ET 
+cos 5 {a —y—Kg(w—y)} cos; {Kg cos yr(w—y)} 
—sin 5 {a°—y? —Kd( u—y) )} sin {K¢ cos (vw — y}], 
and integrate with respect to ¢ cos y, then the final form is 
J, {FKow+y) } 
ie { dy [cosh {2° +y'—K@(e+y)} 
7 KO(e+y) 
e 0 
P | J, {2 Kow—y) 
+ cos 5 {2 —y—KP(e—y)} a i. 
. > K®@®(x2—y) 
