at a Twin Plane of a Crystal, 243 



dielectric medium. If, however, these conditions are fulfilled, 



<W + dQ + dR =0 

 dx dy dz 



P, Q, R being proportional to/, g, h ; and the equations then 

 assume a specialty simple form. 



The boundary conditions which must be satisfied at the 

 transition from one homogeneous medium to another are ob- 

 tained without difficulty from the differential equations. We 

 will suppose that the surface of transition is the plane x = 0. 

 The first condition follows immediately from (2). It is that 

 /'must be continuous across the surface # = 0. Equation (7) 

 shows that dQ/dy + dR/dz must be continuous. From the 

 similar equation in g, viz. : — 



. d 2 g _ d da _ d do 

 dt 2 dz dt dx dt 



V ^ dy\ dx^ dy +dlf' * " ' (8) 



we see not only that dc/dt, or c, must be continuous, but also 

 that Q must be continuous. In like manner from the corre- 

 sponding equation in h it follows that R and b must be con- 

 tinuous. The continuity of Q and R secures that of dQ/dy + 

 dll/dz ; so that it is sufficient to provide for the continuity of 

 A Q, R, b, c (A)* 



Isotropic Reflexion. 



If both media are isotropic the problem of reflexion of plane 

 waves is readily solved. When the electric displacements are 

 perpendicular to the plane of incidence {xy),f and g vanish, 

 while h and the other remaining functions are independent of z. 

 The only boundary conditions requiring attention are that R 

 and b should be continuous, or by (6) that R and dR/dx 

 should be continuous. This leads, as is well known, to 

 Fresnel's sine-formula as the expression for the reflected wave. 



"When the electric displacements are in the plane of incidence, 

 h=Q, and (as before) all the remaining functions are inde- 

 pendent of z. As an introduction to the more difficult 

 investigation before us, it may be well to give a sketch of the 

 solution for this case. In the upper medium we have as the 



* Of these conditions the first is really superfluous. If we differentiate 

 (7) &c. with respect to x, y, z respectively and add, we see that the truth 

 of (2) is involved. In some cases it would shorten the analytical expres- 

 sions if we took T, Q, R as fundamental variables, in place off, q, h. 



R2 



