STUDY OF ORES AND METALS. 



411 



vi _ V2 _ I Sin t _ sin r _ I 



2i h J ?n 2 / 



in which i is the angle of incidence, rangle^ of refraction, go velocity of light in 

 air, q wave normal velocity in the crystal medium and / a constant, real 

 number. Since "q is complex, r is also complex. By means of these values 

 equations ii and 12a can be transformed to read 

 [022 + (022 — /-) tan^ r][dn — 2031 tan r + (033 — /-) tan^ r] 



= {di2 — an tan^ r){i + tan- r) 

 (/^ — 022) tan^ r — dii 



tan 5 



(/2 — 022) tan^ r — a^i 



(13) 

 (14) 



(ai2 - an tan2 7) Vi + tan^ r (/' - ^ss) tan^ r + 2a3i tan r - an ' 



In these equations "ou • • •, r and 5 are complex quantities. A complex 

 value of "signifies that the amplitude in the wave is not constant; a complex 

 value of 5, that there is a phase difference between the amplitudes of the 

 components normal and parallel to the plane of incidence, hence the vibration 

 is elliptic in form. 



For the case assumed, namely, a cr3'stal plate surrounded by air, the 

 boundary conditions (ri)i := (j«)o, {v~)i^{v)2, (w)i=:(w);, (<9X/c)0i = 

 {dX/dt)o can be written by virtue of equations (9), (5a), (6), and (7). 



{E cos € — R cos p) cos i = Di cos 61 cos ?i + D2 cos 52 cos ?2, 



E sin e + i? sin p = Di sin 5i + D2 sin 52, 

 (E cos e + i? cos 5) sin i = Di cos 5i sin ?i + D2 cos 62 sin rt, 

 (E sin e — R sin p) sin j-cos i 

 — sin ri 



(15) 



= Di 



2r 



[sin 5i(aii cos ri — ai3 sin n) + ctia cos 5i] 



+ D2 — ^ [sin 52(011 cos ?'2 — Ci5 sin r2) + ai2 cos 52], ^ 



in which £, R, Di, Dn are the amplitudes of the incident, reflected, and two 

 refracted waves respectively; e, p, S, 5,, the polarization azimuths; q„, Qo, qi, ^2, 

 the normal velocities of the waves, i, — J, ~i, ^2, the angles between the wave 

 normals and the plate normal. 



The last equation of (15) may also be written 



(E sin e — R sin p) sin i cost ^ D^ sin ri(cos r, sin 5 + sin r^ tan s^) 

 + £>2 sin7o(cosn, sin 5^ + sin n tan To), 



(15^) 



wherein 7;^ is the angle between the refracted wave normal and its ray direc- 

 tion. In this form the equation is more convenient for use in certain com- 

 putations. In these equations i, E, e, q^ of the incident wave are known ; i, 

 qa of the reflected wave; also by computation from (12) and (13) r^, n and 

 5j, S^^of the refracted waves; unknown are R, T oi the reflected wave and 

 Di, D„ of the refracted waves. 



1 A dash above a letter is used to signify that the quantity represesented 

 may be complex. 



