Mr. J. H. Pratt on the Supernumerary Bows in the Rainbow. 85 



at the same angle O^—^y 



= ^2 sin </>! - sin ^ <^, jr ^^ , 



substituting from art. 5, 



=0-153 xr/S. 



Let \ be the length of an undulation of red light, and suppose 

 the above interval is m of these undulations, 



/. m.X=0-153xrA 



m .\ 



2r 

 or turning arcs into angles, 



m . \ 

 "27* 



.-. ^=13x 

 angles, 



= 745°x 



Now X= 0*0000266 of an inch. And if the diameter of the 

 drop of rain equal (for example) about 3000 of these undulations, 

 that is, about -^^th of an inch, then 



/o ^° 1 



P=—r nearly ; 



or, putting m=l, 2, 3 . . . . successively, the angular distance of 

 the successive red bows formed by the interference in question 

 from the primary red bow will be 



15^ 30', 45', r, 1°15'.... 



These bows will be of a diminishing intensity, and none of them 

 will have anything like the intensity of the primary red bow, for 

 which m = 0. 



In the same way the other colours — the orange, yellow, green, 

 blue, indigo, and violet — will have their sets of bows of inter- 

 ference at somewhat less and less intervals than those of the red. 

 These bows of interference will not be visible in the *space occu- 

 pied by the primary bow, its intensity being so great as to drown 

 their feebler light ; they may have the effect of somewhat, though 

 perhaps almost insensibly, diluting the purity of the prismatic 

 colours of the primary bow after the red, which colour alone will 

 be absolutely undiluted. But within the violet of the primary 

 bow these bows of interference will begin to show themselves : 

 in this case ^ = 2° and more: hence m = 8 and more, and the 

 bows of interference of the various colours will begin to overlap, 

 so as to produce the general effect of alternating red and green, 

 as seen in Newton^s rings. These are the prevailing colours in 

 the supernumerary bows seen in the heavens ; and we may there- 

 fore conclude that they arise, as here described, from interference. 



The distances of the bows of interference, it will be seen, de- 



