EEC 



Hence the colours in the two bows lie in a contrary order, the reti 

 forming- the exterior ring of the primary, and the interior ring of tlte 

 secondary bow. 



5. To find the altitude and breadth of the rainbow. 



In the primary, the altitude of the highest point cf the red arc = 42. 

 2' sun's altitude j and of the violet 40. 16' sun's altitude. Hence 

 the breadth of the bow, supposing the sun a point = 1. 46' ; this breadth 

 must, however, be increased by 30' the sun's apparent diameter, and .*. 

 the true breadth = 2<>. 1&. 



In the same manner it may be shewn that the altitude of the highest 

 point of the secondary = 54. 10' sun's altitude; and breadth = 3. 42'. 



Cor. When the sun is in the horizon, the altitude of the bow is equal 

 to its radius j if the sun's altitude equal or exceed 42. 2', there can be no 

 primary bow j and if it equal or exceed MO. 10', there can be no secon- 

 dary. 



6. Given the radius of an arc of any colour in the primary rainbow, to 

 find the ratio of the sine of incidence to the sine of refraction, when rays 

 of that colour pass out of air into water. 



The ra dius of the arc" 4 0' 2 q> ; let the tangent of 2 $' <*, half 

 tlu's angle, be a, the tangent of $' ; then 

 2 z 3 3 a z* a o. 



The value of z being thus obtained, the angles $' and and conse- 

 quently their sines may be found from the tables. 



RECIPROCALS of numbers. See Involution. 



RECIPROCAL Spiral See Spiral 



RECTIFICATION of Curves. 



Let z = curve, x and y the abscissa and ordinate ; then 



Ex. 1. In the semicubical parabola, where a x* = y* t 



- ( 9 ff-M)4- Sa 



27 | " 27 ' 



2. In the common parabola, = =- x (y* + b* y*) 9 + I * 



824 



