Reflection and Refraction of Light. 167 



in the values of C,/C, A,/ A, and making M = 0, we get at once 

 Cauchy's well-known formulae. 



Making these same substitutions in (11), we get 



cot j3 A y-_ sin 2 i + cos i UV^-" 1 . 

 cot a sin 2 V— cosiUe M ^ -1 



whence 



A 2Usin u cost sin 2 i . , / a . . Ucosix 



tan A = — t-t-. — rYg s-^- = sin u tan 2 tan -1 — r-^-r ), 



sm 4 i — Lr cos 2 1 \ sin^ i / 



cot 2 /3 sin 4 1 + cos 2 i U 2 — 2 sin 2 i cos i U cos u 

 cot 2 a sin 4 i + cos 2 i U 2 -f 2 sin' 2 i cos i U cos ?4 



= cot(^/r— 45°), 

 where 



cot ilr = cos m sin ( 2 tan~ ! . . ), 



T \ sm 2 1 J 



or, if a = 45°, 



cot 2/8= cos w ( sin 2 tan -1 . . ). 

 V sm 2 t J 



At the polarizing angle I, for which A = 7r/2, we have 



U = tan I sin I, u = 2/3, 



where /5 is the azimuth of the reflected vibrations, when the 

 incident vibrations are in an azimuth 45° with respect to the 

 plane of incidence. 



These values substituted in equations (12) give the values 

 of the constants 6, e, and then these same equations serve for 

 the determination of u, U for any other angle of incidence. 



While the above equations can at the best be only considered 

 incomplete, objections have also been made to the complex 

 value of the refractive index involved in them. 



Lord Rayleigh's criticism* that the real part of ^ 2 should 

 be positive, while the results of experiment substituted in 

 Cauchy's equations give a value of yu, 2 with its real part nega- 

 tive, seems not so much an argument against Cauchy's idea, 

 as an " argument against the attempt to account for the 

 effects on a purely elastic solid theory " f . 



The value of y? resulting from Sir W. Thomsons theory of 

 light is a real negative quantity ; this value substituted in 



* Phil. Mag. [4] xliii. p. 325. 



f Eisenlohr, Wied. Ann. i. p. 204 ; Glazebrook, Brit. Assoc. Report, 

 1885, p. 197. 



