Reflection and Refraction of Ligld. 163 



Hence, if i>I, the value of a! becomes imaginary, and the 

 refracted ray will die out as it leaves the refracting surface. 

 Writing 



U = sin* {i—l) sin (i + I), 



we must substitute in the formulae obtained above 



A 



the negative sign being taken, as the second medium is on the 

 side of negative x. 



Substituting this value, we find that the reflection is total 

 both for the vibration in the plane of incidence, and for the 

 vibration perpendicular to the plane of incidence, and for the 

 difference of phase between the components of the reflected 

 ray we get from (11) 



cot/3 (S _ 5) y-! _ sin i (sin i + UM) 4- cos i (M sin i + U) v^^I _ 

 cota " ~ sini(sini+UM) — cosz(Msini + U)\/^I' 



whence 



$, — & _ cos i { M sin i + sin^ (i — I) sin * (i + 1) } 

 2 sin i {sin i + M sin* (i — I) sin* (i + 1) } 



M^?— + sm*(*— I)sin*(i + I) 



sini v y v / 



= i-s- m . COS I, 



if the square and higher powers of the small quantity M are 

 neglected. 



Cauchy has sin 2 i instead of sin 2 1 in the numerator of the 

 last expression ; the correct formula was first given by 

 Beer*. 



iyvr 



Before proceeding further, it will be as well to discuss the 

 value of the expression denoted above by M. 



Cauchy, not seeing that his equations of condition involved 

 the assumption of the identity of the statical properties of the 

 aether in the two media, adopted tha following relations, 



m + n = — e 2 n, m! + n f = — 6 !2 n'j 



where e, e' are very small numerics. 



* Pogg. Ann. xci. p. 274. t C R. ix. pp. 691, 727. 



