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



spreads in the plane of the paper through the space 

 /, . de\ D d0 



This is reduced to w when the deviation is a minimum, and the 

 pencil emerges parallel in the plane of the paper. 



11. From the last two articles, it appears that the ratio of the 

 spaces occupied by the emergent pencil at a distance D and by 

 the incident pencil 



^ D sin^ f-^ + fl -f 5 sec 6)-\ 

 r sin</) L \ r ^ ) d^j' 



Put ^1— /S and 4>]^oi for 6 and (j)j and reduce, observing that 

 6 1 and (^1 satisfy equations (1) and (2), and neglecting higher 

 powers of a than the first, and therefore /3, and we have the 

 above ratio 



- D sin e, f / 2+sin^<^i D 3sinj>i \ "\ ^ 

 ~" r sin </>! I \ sin 2(/)i r 1 + cos 2^i / J ' 



or, neglecting the first term within the second brackets as being 

 extremely smaU compared with the next, 



_ D sin <9i D^ 3sin<9i 



"~ r sin (pi r^ 1 + cos 2</)i ' 



or substituting the values given in art. 5, 



18 D 4D2 



= 237 + 7^"- 



12. The intensity of the light which comes to the eye varies 

 as the reciprocal of this expression. A quantity of light is lost 

 in the refractions and reflexion, which is some function of the 

 angle of incidence : call it ¥{^^—u). Then the intensity of the 

 hght which comes to the eye 



When a=0, that is for the primary bow itself, the intensity of 

 the light 



=^(*i)-l8D- 



But for the supernumerary bows, in which u is not less than ^, 



the first term within the bracket is incomparably smaller than 

 the second. Hence for these bows the intensity 



