162 Mr. J. Walker on Cauchy's Theory of 



Hence 

 2A=/* 2 + - + -(> 2 --l)MV^l 



a 2 + o 2 a a a z + b z 



a(a 2 + 6 2 ) (a+a;-^±l6 , say, 



( fl a'-6«)-ft(a' + a)M^=i ^ ,v 2E ^=i sav 

 JA,- a( y + ^ • ( a -<>>)- &*>fi , sa 7- 



Then R, R / denote the amplitudes of the incident and 

 reflected vibrations, and 8, S / the difference of phase between 

 the incident and refracted, the reflected and refracted waves 

 respectively. 



Hence, if a is the azimuth with respect to the plane of inci- 

 dence of the incident vibration, the reflected vibration will in 

 general be elliptical with a difference of phase S y — $ between 

 the components in and perpendicular to the plane of incidence ; 

 and if this difference of phase is destroyed, the azimuth /3 of 

 the resulting rectilinear vibration will be given by 



cot/3 = R/C y . 

 Hence 



cota (aa' + & 2 )+fc(a'-a)MV_l' " ^ ; 



and 



cot 2 /3 _ (flq'-& 2 ) 2 + ftV + a) 2 M 2 

 cot 2 a ~ (aa' + & 2 ) 2 + £ 2 (a'-a) 2 M 2 



cos 2 (j + r) + M 2 sin 2 (i + r) m 



cos 2 (i— r) + M 2 sin 2 (i — r) ' 

 also 



, ,* j,v Mjtan (i + r) + tan(z— r)} 

 tan(d, 6)- 1 _ Wt . dn ( i + r ) t r in{i _ r y 



Total Reflection*. 

 If fj, is less than unity, we may write yu- = sin I, and we 



4tt 2 



'' 2 ^- -, T cos 2 r= -^-sin (I — i) sin (I + i). 



* C. R. ix. p. 764 ; xxx. p. 465. 



