for Reflected and Refracted Light, 13 



before, if we assume k^=2hi to find k and k', the condition of 

 equivalent vibrations (L) gives 



, ,, , cosr . . .cosr ^ . ^ 



k + k'=k. r=48inrcosz : = 2sin3r. 



'cosz cos 2 



Also the equation of vis viva (M) becomes 



(A:2-F)=^>-^=4sin2isin2r==(it + ^')(^-^'). 



whence 



k-k'=2sm2i 



2k = sm2r+ sin2i 



2k' = sin 2r — sin 2i. 

 Hence, as before, 



,,__ sin 2r— sin 2i , __ 4 sin rcos i .^. 



~" sin2r-f8in22' '"^ sin2r+sin2e* ' * ^ ' 



And since for i='m or (« + r) =90°, sin 2r=sin 2«, we have A;'=0; 

 for incidences, i < ct, we have sin 2r < sin 2i, and therefore — A;'; 

 and for i > ot sin 2r > sin 2z or + k', 



44. Or again, without assuming the value of k^, we may proceed 

 thus : from the equation of vis viva, 



' sin z cosr ' * 

 from the law of equivalence, 



^ ' cosV 

 Equating these, we have 



(F — ^'^) sin r cos^ r cos z = (Z: + ^^)^ sin z cos^ i cos r 



(F-P) sin 2r= (^ + yt')2 sin 2i 



F(sin 2r - sin 2i) = 2kk' sin 22 + ^^(sin 2% 4- sin 2r) j 



and observing that 



2M' sin 2i=kU(^ (sin 2?— sin 2r) + (sin 2i + sin 2r) ) 



(A; + A;')^(sin 2r — sin 2f) = (A; + A:')^'(sin 2 i + sin 2r) 



A: (sin 2r — sin 2i) = ^'(sin 2r + sin 2z) 



cos z 



A: + A:' = 2 sin 2r, and thence k{=-2 sin 2r ; 



' ' cosr 



or as before, dividing by k, 



J __ sin 2r — sin 2i t. _ 4 sin r cos i 



sin 2r + sin 2i' ' sin 2r -|- sin 2i 

 45. Now it is to be observed that these formulas differ from 



