14 Prof. Powell on the Demonstration of FresnePs Formulas 



those of Fresnel, as before given, in that the numerator of ^ is 

 here negative for t < to- and positive for t > w. 



This difference is traceable to the process by which the deduc- 

 tion of Fresnel's original form was effected ; it being made by 

 means of two equations (see Airy's Tract, § 129), one of which 

 is the same as the equation of vis viva, viz. 



sin r cos i{k^ — A/^) = sin t cos rk^^ ; 

 the other is 



cos i{k — ^) = cos r k^, 



which involves a different assumption as to the principle of equi- 

 valent vibrations, as before observed (30). 



46. It will be desirable to follow the deduction of the for- 

 mulas on this supposition. 



For vibrations perpendicular to the plane of incidence, from 

 the vis viva, 



smrco.i 

 ^ ^smicosr ' 



From this law of equivalence (30), 



{h-h')^=h,^. 

 Hence 



{h^—h'^) sin r cos « = [h—h^Y sin i cos r, 

 or 



{h + A') sin r cos i = (A — A') sin i cos r 



A(sin i cos r — sin r cos i) = A'(sin r cos i + sin i cos r), 

 whence 



y__ sin (i— r) _ 2 sin r cos e 

 8in(2 4-r) ' sm(«+r) 



These differ from Fresnel's original formulas in the sign of A'. 



47. For vibrations parallel to the plane of incidence, on the 

 same supposition (30) we have 



'cos I 

 and thence, as in (44), 



{k'-k!^ sin2r=(^-;fc')^ sin 2« ; 

 whence, in like manner, 



(A:— A:')A:(sin2i-sin2r)=:(^~^)^(sin2i-f sin 2r), 

 and thence 



^ sin 2i - sin 2r , , 4 sin r cos i 

 sm 2t -f sm 2r' ' sm 2? h- sm 2r' 



