Mr. J. H. Pratt on the Supernumerary Bows in the Rainbow, 79 



4. When tlie rays emerge as well as enter parallel, 6 remains 

 constant for that pencil while <j> varies ; hence by differentiating 

 (1) with respect to </>, 



cos<^=^//,cos(^^(9+ ^</)). ... (2) 



values of <f) and 6 

 aares of (1) and ( 



Let <^, and 6^ be the values of <f) and 6 which satisfy these equa- 

 tions. Adding the squares of (1) and (2) and reducing, we have 



Then (1) gives 



These two formulae give 



^1 = 59° 32', e^ = 42° 24' for red rays ; 



<^i = 58° 46', 0^ = 40° 30' for violet rays. 

 The breadth of the primary bow is therefore = 1° 54'. 



5. In the following calculations only the red rays will be con- 

 sidered j and we shall find the following values useful : 



sin- <^i = -496469, sin ^i= -861924. 



tan<^i = l-7, cos 2(^1= -0-485827. 

 sin6>i = -674517. 



6. The supernumerary bows are always seen close to the inner 

 limit of the primary bow ; and 6^ — 6, or the angular distance of 

 any point in them from the red of the primary bow, varies from 

 2° to 4°, or reducing angles to arcs, from ^^^^th to y^^-th part of 

 the radius. 



To find how 6 and <f> vary together for points within the limits 

 of the supernumerary bows, let ^ = ^i — /8 and ^ = ^j — a ; a and 

 P will both be small. Substitute these values in (1), 



.-. sin(</>,-a)=/isin (^^6'iH-g</)i~2^-^«j. 



Expand in powers of a and ft and reduce, observing that 6^ and 

 <^i satisfy equations (1) and (2) ; the result is 



anan(^^--^=^-l-^ ^^ tanc^i^ ^ ^^ ^ 



For a first approximation neglect ^ and a^ . . . . , then 



^= — tan<^i .a^= ja*, -/ tan</>i=:l-7. 



For a second approximation retain a^, but reject a'*, and there- 

 fore ^^ -J reduce and transpose, divide by the coefficient of a^, 

 and put tan <^j = l-7 ; then 



