‘ 
; 
Rainbow due to a Circular Source of Light. 607 
supernumerary bows due to the sun, white colour predo- 
minates, and we cannot distinguish many numbers of the 
supernumerary bows. This explains the fact that the rainbow 
in nature is accompanied by a small number of supernumerary 
bows only, while according to Airy’s theory the rainbow ought 
to be accompanied by numerous bows. 
According to § 5, we notice that the difference between the 
maximum and minimum values of intensity becomes larger 
with the size of the drop for a point source ; but for a circular 
source the intensity depends on two factors, one of which 
enjoys the same property as for a point source, but the other 
produces a contrary effect. Montigny* says that supernu- 
merary bows are numerous when the drops are small. This 
holds for the case of a circular source and supports our view, 
but he considers this as the result of Airy’s theory, 7. e. of a 
point source, which we cannot understand. 
9. As the consequence of the above discussion, we obtain 
the following result, where (1) represents the case of a point 
source, (iI) a circular source:— 
(a) The positions of the maxima and minima of (II) 
approximately coincide with that of (I) (the first maximum 
of (II) is displaced by a small amount towards @=0 as 
compared with (I), and for other maxima and minima this 
displacement becomes smaller and smaller). But the maxima 
of (II) may correspond to the minima of (1), and the minima 
to the maxima. 
(6) The value of (IL), which corresponds to the maximum 
of (I), is smaller than that of (I), and the minimum value 
of (II) is greater than that of (1). This difference between 
(1) and (II) becomes larger with the larger value of ® (2. e. 
with the larger diameter of the source). 
(c) As the value of @ increases, the maximum value of (1) 
and (II) gradually decreases, while the minimum value of 
(I) always remains 0; but the minimum value of (II) 
gradually increases until it becomes equal to the maximum 
value and assumes a stationary value, then the maxima and 
minima interchange, the difference at first increases and 
then decreases, then again assumes a stationary value, and 
soon. Ifin this interval between the two stationary values 
‘ the maxima of (II) correspond to that of (1), then in the 
next interval the maxima of (II) correspond to the minima 
of (1). 
(d) For larger values of 7 (radius of the drop) the intensity 
* Phil. Mag. ix. p. 389 (1880). 
2U 2 
