Theory of the Rainbow. 599 
suggests itself in conjunction with the problem of the circular 
source of light—if we take account of the breadth of the slit, 
assuming its length to be infinite, what difference will occur ? 
This must be answered. In the following, we shall start by 
briefly stating Airy’s theory, then proceed to find differences 
when the source of light is replaced by a small circular disk ; 
and after some additional note for the two-dimensional case, 
experimental results will be discussed ; and lastly the colours 
of the rainbow due to the sun are calculated in one case, 
which may be taken as an illustration of the difference between 
the point and the circular source. 
2. It will be necessary, in the first place, to state Airy’s 
theory in a form convenient for subsequent investigation. 
Describe a unit sphere having the centre C coinciding with 
that of a raindrop, and let a point O on the sphere be the place 
of observation, and § the direction of the sun supposed to be 
a point as seen from C. The position of the observer with 
respect to the sun is specified by the angie SCO, or by the 
angle @, which is equal to the angle of minimum deviation D 
minus SCO. 
Put r=radius of the drop, n=index of refraction, 
p—1=number of internal reflexions, 
and ae pn 
eC are 
then the intensity of light at O is given by 
OS Be aa 
(0) = const. ‘ea F788), 
6\3/7 \F 
K=2(;) oe 
f(K@) =| cos > (wW— KOu)du. 
0 
where 
Airy expanded /(K@) as a power series of K@, which is not 
convenient for practical calculation for values of K@ >3, though 
it always remains convergent. On the other hand, especially 
for larger values of 6, the following semiconvergent series due 
to Stokes * can be employed with advantage :— 
* Collected Papers, ii. p. 329 (London, 1883). 
