1888. ] Observations of Supernumerary Rainbows. 283 
Thus sin d =msin fd’ 
(n+1)cosh=pcos qd’ )’ 
which determine the position of the geometrical bow. 
dp dD 
Now 4? =acos/ Ge 
T= — (asin oa +4 008 6538) [ (Ga) 5 
and r is very great, so that 
-1 @#D 
Bience " ~ (a cos op)” dp* 
_2(n+1) ae’ 
~ (acos )* d¢” 
_ 2(n+1) wsin¢’(n+1)*—sin 
(a cos )” po cos b 
_— _ 2n(n+2)sin¢d 
a’ (n+1) cos’ 
aria sees a : 
We pal Na ei (w?—1)° P 
which gives the value of b/. 
The observations of Prof. Miller relate to the cases n= 1, n= 2, 
the primary and secondary bows. 
Although the formule here given apply strictly only to the 
bands whose angular deviation from the geometrical bow is not 
considerable, it has been thought well to make the comparison 
with observation through a considerable range of angle. 
I owe to Prof. Stokes the remark that, at a sufficiently great 
angular distance from the principal bow, the interval between 
successive bands may be calculated simply from the interference 
of the two effective rays, as in Young’s original aper¢u. 
We first examine the primary bows. Applying the formule just 
* This result was given in a question in the Mathematical Tripos, June 2, 1888 
(Camb. Univ. Exam. Papers, 1888, p. 560), I find that the same expression is 
given in the Comptes Rendus, May 28, 1888, by M. Boitel. See also Philosophical 
Magazine, Aug. 1888, p. 239, 
