at a Twin Plane of a Crystal. 245 



when #=0. In the lower medium, when a=0, 



so that by ( 1 4) the continuity of dc/dt leads to the same con- 

 dition as the continuity of/. 



The usual formula for the reflected wave is readily obtained 

 from (15), (16). If 0, fa be the angles of incidence and re- 

 fraction, 



Y 1 2 /V 2 = sin 2 fa/sin 2 fa 



Pilp= (Pi/q)-*~{l>/q) =cot cfr/cot ; 

 so that 



1 — 6 1 _ sm 2 fa cot fa _ sin 2 fa 

 1 + 6 sin 2 cot sin 20 

 Accordingly, 



g /== sin 20— sin 20! _ tan (0-0Q _ ^ «« 



sin 20-+- sin 20! tan(0-|-0i)' ' 



The insertion of this value of #' in (12) gives the expression 

 for the reflected wave corresponding to the incident wave 

 (11). The ratio of amplitudes in the two cases, being pro- 

 portional to \/ (f 2 +g 2 ), is represented by 6', and (17) is the 

 well-known tangent-formula of Fresnel. 



Propagation in a Crystal. 



In a homogeneous crystalline medium, the relation of force 

 to strain may be expressed 



P, Q, R=4*-(<*.Y, h'g, tfh) . . . (18) 



where a ly fr b C\ are the principal wave- velocities. We here 

 suppose that the axes of co-ordinates are chosen so as to be 

 parallel to the principal axes of the crystal. The introduction 

 of these relations into (7), &c, gives 



cPf/dt* = a 2 \/ 2 f-dUjdx^ 



J^dl^b^g-dn/dyL . . . .(19) 



d 2 / l [dt 2 = c 1 2 \/ 2 h-dU/dz) 

 where 



Il = a l 2 df/dx + b ] 2 dg/dy + c 1 2 dh/dz . . . (20) 



The principal problem of doable refraction is the investiga- 

 tion of the form of the wave-surface. By means of (19) we 

 can readily determine the law of velocity (V) for various direc- 

 tions of wave-front (I, m, n). For this purpose we assume 



/, 9, >' = (\ /*, ►)«"*, (21) 



