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



Putting —a' for a, and disregarding the change of sign in the 

 intensity because it arises only from the crossing of the rays 

 within the drop, the intensity of the other pencil of light which 

 emerges with the same angle of deviation as the one above con- 

 sidered, and mixes with it. 



And therefore the ratio of the first to the second 

 _ F(^i-«) ^ 



As ^1— a and </>! + «' do not differ much (being about 54° 30' 

 and 64° SCV), I assume that the first factor of the above = 1 ; it 

 is, in fact, somewhat greater than unity. Hence the ratio of 

 intensities 



a 8- \/50 4 



13 



= r— nearly, close to the primary bow. 



This ratio departs more and more from unity, though slowly, 

 in passing from the inner limit of the primary bow. The factor 

 put = 1 departs from unity very rapidly ; because as <f) increases, 

 the quantity of light lost by external reflexion increases rapidly. 



From this it appears, that the two portions of light emerging 

 with the same deviation are, for points within one or two degrees 

 of the primary bow, of intensities sufficiently near each other to 

 allow of their interfering ; though, because the intensities are 

 not exactly the same, the interference will not be total ; but there 

 will be rings of alternately strong and feeble intensity. As far, 

 then, as intensity of light is concerned, the supernumerary bows 

 may arise from interference. We must now see whether the dif'- 

 ference of paths will agree with the same. 



13. From A (see fig. 1) draw AP and AQ perpendicular to 

 the incident and emergent rays. The space Pa6cQ, estimated 

 by the motion of light in air, ^2 .Ta + ^fi.ab, 



=4rsin(^</>+ -(9)sin(^(/)- ^6^ +4r/^cos (^ ^+ 4^) 



= 2rcos- ^— 2rcos^</) + 4/Arcos (n0+ T^)- 



Put, first <^i — a and 0^-/3, and then </>! + «' and ^,— /9 for 

 <f) and Of and subtract and reduce, neglecting powers of a and a! 

 higher than the first, and therefore ^ altogether : the result gives 

 the difference of paths of the two streams of light which emerge 



