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XV. The Supernumerary Bows in the Rainbow arise from Inter' 

 ference. By Z, H. Pratt, Esq. 



To the Editors of the Philosophical Magazine and Journal. 

 Ge ntlemen, 

 1. rriHE question having been started, '' Do the supernume- 

 A raiy bows in the rainbow arise from interference ?^' — not 

 knowing any book where the doubt was resolved, I set to work to 

 make the necessary calculations; and as they may be of interest 

 to your readers, I forward them to you for publication, should 

 you see fit to give them a place in your Magazine. 



2. Let r = rad. of a drop of rain. 

 fi = index of refraction for water. 



= 1*331 for red, 1*344 for violet rays. 



<f> = angle of incidence of a ray. 



(f>'= angle of refraction, sin </)=/x. sin </>'. 



6 = angle of inclination of an incident ray with the same 

 when it emerges after two refractions and one intermediate 

 reflexion within the drop. 



<^, and Oi the values of </> and 6 corresponding to those rays 

 which emerge parallel. 



D = distance of drop of rain from the spectator. 



3. Let the plane of the paper pass through the sun, the eye 

 of the spectator, and the centre of any given drop of rain ; and 

 let (fig. 1) Sa^cE be the coui'se of any ray from the sun, twice 

 refracted and once internally reflected by the drop. 



Fig. 1. 



It is easily seen that ^=4<^'-.2</> ; putting this value of <^' in 

 sin <^= /i sin ^', we have 



sm 



^=/tsin(ie+i^) (1) 



