83 Mr. J. H. Pratt on the Supernumerary Bows in the Rainbow. 



the intersection SC of the planes. Hence these rays make with 

 each other the angle sin 6 x angle of inclination of the planes 



= ; — r : and therefore at a distance D from the drop, the 



rays are spread, at right angles to the paper, through the space 



sin(^-f<^) Dsin^ 

 sm 9 r sm 9 



The first term may always be neglected with respect to the 

 second. For D must be an enormous number of multiples of r, 

 the radius of a di'op of rain ; and the only case in which the first 

 term might appear to be large and the second small is when 

 0=0, and therefore ^=0. But in this case, (1) in art. 3 

 gives 4i(j>=/JL6 + 2fi(l), and the above expression becomes 



4— u D 4— 2a 



w -\-w —, 



fi r fM ' 



and the first term is always extremely small in comparison of the 

 second. Hence the space through which the pencil spreads at 

 right angles to the paper 



D sin^ 



= W r— r. 



r sm9 



10. Now I will calculate the divergence in the plane of the 

 paper. (See fig. 1.) 



Let SflZ>cE and ^'a't'c'E' be the course of two rays of the inci- 

 dent pencil in the plane of the paper. The arc 



abc= (27r-4<^V = (27r-^-20)r, 



l+^jr.8(/>; '.' aa'=—rh<f>; 

 and the width of the pencil at c 



=cc' cos <^= f 1 + -rr )r cos </). S<^ ; 

 and the width at a (or «;) =r cos </> . 8(/> ; 



.*. width at c^w\\+ -rA. 

 The rays cE, c'E' make an angle h6 with each other, or 



d<j) rcoscf)* 

 and therefore at a distance D from the drop of rain, the pencil 



