STUDY OF ORES AND METALS. 



409 



in the form 



— - = - (au^ + ai2V + aisH, 

 at c 



dY I _ 



-— - = - (a21? + 022?? + a23f), 



at c 



r\7 T 



-rr = - (flsi? + 0321? + Osaf), 

 at c 



(6) 



in which the complex constants (called polarization constants) ahk = akh-\-ibhk 

 are simple determinate functions of c^ and of the complex quantities ^n . . . 

 ^22 . . . ; in these equations dhk = dkh. 



To eliminate the components of the electric force X, Y, Z differentiate 

 equations (5&) after the time 



c dt^ 



dz\dt ) dy\dt ) ' c dt^ dx \dl ) dz\ dt J ' ^'^' 



I d^u _ d /dX\ d /dY\ 

 c dt^ ~ dy\ dt ) dy\dt)' 



and substitute in these equations the values of dX/dt, dY/dt, dZ/dt from 

 equation (6), and obtain the equations 



t;^ = -r- (a2i5 + 0227? + a23f) — -r- (OsiS + azirj + CssD. 

 dr ax ay 



d'^v d . . , , \ d , . , , . 



TTi ~ T' ^^31^ + 0327? + ossf) — -T-{auk + Oi2»7 + aisD, 

 dr ax oz 



^7;r = T- (^11^ + 0,1171 + cijf) - — (a2i$ + a22»7 + 023?), 

 di- dv ax 



(8) 



which are free from the components of the electric force and of the electric 

 current. 



A particular solution of these equations is 



u = mp, V — flip, w = H3p, 

 ~ ' T A ''1^ + v^y + V3Z 



p = Ae 



{'- 



). 



(9) 



in which Mi, ^2, /"s are the direction cosines and A (complex) the amplitude 

 of the polarization vector ; v^^ v„, v^ the direction cosines of the wave normal 

 and g" (complex) the velocity. If in (9) we assume vertical incidence, then 



^j = j,2 = o, i-s = I ; if we put = = , and X = T-q, wherein q, k, and X are 



real, then 



u = niAe 



-2nKli2^(-j.-l) 



(10) 



This equation states that the amplitude of the wave of light after passage 

 through the path X (one wave length) has decreased to the extent of f"^"^'"; 

 K is therefore called the absorption index. 



