608-610] Crystalline Media 529 



By equation (92), the electric energy W per unit volume in such a medium 

 is given by 



n ...... . 



If we transform axes, and take as new axes of reference the principal axes 

 of the quadric 



K u a? + 2K u xy+ ...... = 1 ..................... (592), 



then the energy per unit volume becomes 



W = ~ (K,X* + K, F 2 + K,Z>\ 



oTT 



where K lt K 2 , K 3 are the coefficients which occur in the equation of the 

 quadric (592) when referred to its principal axes. The components of polari- 

 sation are now given by (cf. equations (89)), 



47r/ = K, X, 47r<7 = ^ 2 F, Mi = K S Z. 

 The equations of propagation (putting p 1) now become 



= _ = ____ 



C dt ~~dy dz C dt dy 'dz 



K*dY_-fa__ty\ ld^_d^_dZ 



~G ~dt " dz dx f ' C~dt~dz dx 



K 1 dZ = d/3_da\ 1^I = ^Z_^ 



G dt ~ dx dy) C dt ~ dx dy 



If we differentiate the first system of equations with respect to the time, 



and substitute the values of -yr , -yr , ~- from the second system as before, 



dt at at 



we obtain 



- 



l a \ X ^ + ^- + ^- , etc. 

 dt' dx\dx dy dzj 



On assuming a solution in which X, F, Z are proportional to 



e u(lx+my+nz-Vt) 



these equations become 



K,X = X-l(lX + mF-f nZ) = 0, etc. 



On eliminating X, Y and Z from these three equations, we obtain 



8 * 



4 



O 2 

 If we put -j=- = vf, etc., and simplify this becomes 



This equation gives the velocity of propagation Fin terms of the direction- 

 cosines I, m, n of the normal to the wave-front. The equation is identical 



j, 34 



