STUDY OF ORES AND METALS. 415 



031 = 031 +^'&3i^(^^ — ^^) cos 6 sin 6, 

 a^o = ^12 + '^12 = O- 



In these equations a^ • • • and a", &, c are complex quantities. On 

 substituting these values in equations (10'), (ii'a) and (15) we find 



(q^ — <2ii)(2" — 022) =0 or §2" = fln Si" = ^22, 



TT 



5i = o, ^2 = -, 



TT 



ei = O, €2 = ~ . 



2 



The refracted waves 8^, So are thus plane-polarized. Equations (18) 

 reduce to the form 



Rs _ RsiEp2 _ Rsi __qo — Qi 

 Es Ep^Esi Esi 2o + er 



But from equation (10) 



ii ^ gi ^ I . 



go 2o(i — in) Wi(i — •iK) ' 



accordingly 



Rs ni — I — ifiiKx 

 Es «] + I — iiiiKi ' 



The right-hand side of this equation is a complex quantity and of 

 the form 



a + ib aA ^-hB + i{bA - aB) 

 A +iB ~ A'^-\-B'' - / + *^- 



This in turn may be considered equivalent to the expression 



{P-\-iQ)e'^=iP + iQ) (cos T-j-isin r) 



= P cos T — Q sin T-|- i(P sin t + Q cos t). 

 If now 



P^r cos A, 



Q ^ — r sin A, 



the amplitude may be written for £.?= i 



Rs = r-^ cos (r — Aj-f rr^ sin (r — l^^)^f -\-ig = r^e ''~^' (21) 



(20) 



