The Philosophical magazine; a journal of theoretical, experimental and applied physics . divide by the coefficient of a.^,and put tan 0i = 1 7; then 80 Mr. J. H. Pratt on the Supernumerary Bows in the the first approximate value of or in the second term. 1 4 ? j3, .•/S^ is neglected, 2 , 1 ^ ••.«=±5 ^5^+4/5. Call these two values a and —a. 7. This shows that for every value of /S there are two valuesof a, one positive and the other negative, and nearly equal toeach other; that is, there are always two values of 0, one lessthan ^1 and the other greater, which give the same deviation
The Philosophical magazine; a journal of theoretical, experimental and applied physics . divide by the coefficient of a.^,and put tan 0i = 1 7; then 80 Mr. J. H. Pratt on the Supernumerary Bows in the the first approximate value of or in the second term. 1 4 ? j3, .•/S^ is neglected, 2 , 1 ^ ••.«=±5 ^5^+4/5. Call these two values a and —a. 7. This shows that for every value of /S there are two valuesof a, one positive and the other negative, and nearly equal toeach other; that is, there are always two values of 0, one lessthan ^1 and the other greater, which give the same deviation^1 —/8; and therefore the corresponding rays emerge parallel toeach other. It is the interference of these rays, which take dif-ferent courses in the drop of rain and emerge parallel, whichcauses the supernumerary bows. This I proceed to show. The manner in which these rays pass through the drop isshown in the accompanying figure (fig. 2). The dark line inthe middle is the ray of least deviation, cE and cE emergeparallel, and the pencils mix on reaching the eye. Fig. Mr. J. H. Pratt on the Supernumerary Bows in the Rainbow. 81 8. I will first find the intensities of these emergent rays, andcompare them. A cylindrical pencil of parallel rays falling on the diop ofrain will emerge through an element of the surface of the drop,of dimensions difierent to those of the element through which itpassed on entering; and the pencil will emerge, in the generalcase, not parallel, but diverging, both in the plane of the paperand at right angles to that plane. It will be necessary to cal-culate this in order to find the intensity of the emerging light. 9. And first, I will calculate the divergence at right angles tothe paper. Fig. 3.
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