for Reflected and Refracted Light, ' 15 



which are the formulas of Fresnel as originally deduced, and in 

 which (as before) we have + A:' for i < «r, and — A:' for i'^ 'ur. 



48. Here, again, we might otherwise deduce the same for- 

 mulas; as before (43) assuming A:y=2Ap we should have 



A:-/t'=/t^=2sin2r, 

 cos I 



and (A;^— A:'^) =4 sin2r sin %i gives A; + A:' = 2 sin 22, whence 



,, sin2z— sin2r , , 4 sin r cos 2 



sm2«+sm2r ' sin22 + sm2r 



49. In another point of view we may observe the relation sub- 

 sisting between all the above formulas (generalizing a remark in 

 Mr. Power's paper), as follows : — 



Trigonometrically, we have evidently, 



sin^(zH-r)— sin^ (i— r)= sin2isin2r, (P) 



[sin 22 + sin 2r] ^ — [sin 2i — sin 2r] ^ = 4 sin 22 sin 2r. (Q) 



50. Also from /^, sin r = sin f, we have the identical equation 



4/A sin r sin i cos^ i cos r = 4 sin^ i cos^ i cos r, 

 or 



. fjb cos 2(sin 2r sin 22) = cos r(sin^ 22) ; 



whence, substituting in this equation the value from (P), it 

 becomes 



/u.cos2[sin^ (2 + /*)— sin^ (i— r)] = cosr(sin2 22) ; 

 or from (Q) we have 



fi cos i[(sin 22 + sin 2r)^— (sin 22— sin 2r)^] = cos r 4 sin^ 2a; 



which agree with the formulas of MaccuUagh, and the equation 

 of vis viva on the hypothesis of equal densities. 



51. Again, we have the identical equation 



4 cos^2' sin 2 sin r cosr = 4/-t sin r cos^ i sin r cos r, 

 or 



cos 2 (sin 22 sin 2r) =^6 cos r(4 cos^ i sin^ r) ; 



whence from (P) we have 



cos i [sin^(2 + r) — sin^(2 —r)] =ft cos y*(4 cos^ i sin^ r) ; 



or from (Q) we have 



cos 2 [(sin 22 + sin 2r)^-- (sin 22— sin 2r)^] =fjb cos r(16 cosSsinV) ; 



which agree with FresnePs formulas, and the equation of vis viva 

 on the hypothesis of increased density. 



52. But thus far nothing determines the plane of vibration. 

 For this purpose we must apply the law of equivalent vibrations ; 

 and in either case, since it is the squares of the amplitudes which 

 enter the formulas, we may take + A', + h^, ± A:', + k^y according 



