FresneVs Reflexion of Light Theory. 307 



When %S = S' (referring to fig. 1), 



^7 (2), % -BA-AC = BD-DC, 



by (1), %M - BA = DO = V(BA)" s +/^-l» 



whence 



v %y-i 



But, taiii=/*/BA, 



wherefore tan i =fi \J % ^ J~" (3) 



Again, (BC) 2 = /^ 2 +(AB) 2 , 



therefore BC = a /ffi~"* . 



v %-y-i 



But, sin t = AC/fiC=^/BC, 



wherefore sin i = //a / ^- ^ ~ (4) 



v x v-i 



Fresnel's formula for the ratio of reflected amplitude is 

 tan (t— r)/ tan (i + r) when the light vibrates in the plane of 

 incidence, and sin (i — r)\ sin (i + r) when the vibration is 

 perpendicular to that plane, and we shall require these ratios 

 expressed in terms of S and S'. Referring to fig. 1 where 

 the angle ABC = f, and the angle BCD = r, 



tan (i — r) sin i cos i — sin r cos r 

 tan (i + r) sin i cos i + sin r cos r 



AOAB-BD-DC 

 " AC-AB + BD-DC* 



But, by (2), AC'AB : BD-DC : : S : S', wherefore 



tan(t-r) _ S-S' 

 tan (i + r) ~ S + S'" 



(5) 



Again, sin(t-r) __ AC'DC-AB-B D 



sin (i + r) ~ AC-DC + AB-BD' 



X2 



