at a Twin Plane of a Crystal, 247 



plane. If one of these angles were to vanish, B would dis- 

 appear, in spite of a powerful double refraction. 



For a general solution of the problem of reflexion from a 

 twin plane, we should have to suppose the plane of incidence 

 to be inclined at an arbitrary angle to the plane of symmetry 

 (.v,y) ; but we may limit ourselves without much loss of 

 interest to the two principal cases, when the plane of in- 

 cidence (1) coincides with the plane of symmetry, (2) is 

 perpendicular to it. 



Incidence in the Plane of Symmetry. 



Under the first head there are two problems which may be 

 considered separately. The simplest is that which arises 

 when the vibrations are perpendicular to the plane of inci- 

 dence, that is, are parallel to z. It is not difficult to see 

 that in this case the difference between the twins never comes 

 into operation, and that accordingly the reflexion vanishes ; 

 but it may be well to apply the general method. 



Since/, (7, and therefore [by (25), (2(5)] P and Q, vanish 

 throughout, while h and R are independent of 2, the two first 

 of equations (7) are satisfied identically,, and the third becomes 



. d 2 h _ d*R d*R 



dt' 2 dx* df 

 or by (25) 



dVi -r. (dVi , dVi\ /a _ 



This equation applies to both media, since there is no change 

 in the value of D. Thus, so far as the equations to be 

 satisfied in the interior are concerned, the incident wave may 

 be supposed to continue its course without alteration. 



It is equally evident that the general boundary conditions 

 are also satisfied. For/, Q, c vanish throughout, and by (6) 

 the continuity of R and b merely requires the continuity of 

 h and dhjdx. Since all the conditions are satisfied by 

 supposing the incident wave to pass on without alteration, it 

 is clear that there can be no reflected wave. 



We have next to consider the case when the vibrations are 

 executed in the plane of incidence, so that h vanishes, while 

 (as before) all the remaining functions are independent of z. 

 On account of the symmetry there can be but one reflected 

 and but one refracted wave, and in each /* must vanish. We 

 may, therefore, take the following expressions as applicable to 

 the various waves : — 



