282 Mr J. Larmor, On Prof. Miller’s [Oct. 29, 
light of index pw is as follows. Obtain the equation of the emerging 
wave-front, which will be of the form 
pz+be+...=0 
in the neighbourhood of the part efficient in the formation of the 
bow. The dark and bright bands correspond respectively to the 
values of m which give the maxima and minima values of the 
expression 
(eo) “| 2 
| | cos — (w*— mw) dw | 
ee 
If m, denote such a value, the angular separation of the correspond- 
ing band from the geometrical bow is y, where 
» denoting the wave-length of the light. 
To make a comparison, it remains therefore only to determine 
the value of b. In the case of nature, when the refracting drop 
is a sphere and the incident beam parallel, this is easily ac- 
complished. 
For the geometrical caustic is the evolute of the wave-front; and 
its radius of curvature p at the bow is easily found to be given by 
=>=— r, 
di 
where r is the radius of curvature of the wave-front, i.e. the 
distance of the caustic from it measured along the ray. Now to 
calculate p; let @ denote the angle of incidence of a ray, ¢’ its 
angle of refraction; a the radius of the drop, and p the perpen- 
dicular from the centre of the drop on the emergent ray whose 
deviation is D. Let us consider the n™ rainbow. We have 
do, el @ 
pP=Pp ain aD® ’ 
where p=asin ¢, 
D=2(¢—$) +n (w—29) 
=nr +26—2(n+1) $; 
and as D is stationary at the bow, . 
dD dq’ 
cag so that aps 
