IN CRYSTALLIZED MEDIA. 301 



If now in our indefinitely extended medium we wish to 

 determine the laws of propagation of plane waves, we must 

 take, to satisfy the last equations, 



u = af (ax + by + cz + et), 

 v = pf(ax + by + cz + et), 

 w =yf(ax + by + cz + et) ; 



a, b, and c being the cosines of the angles which a normal to 

 the wave's front makes with the co-ordinate axes, a, /3, 7 con- 

 stant coefficients, and e the velocity of transmission of a wave 

 perpendicular to its own front, and taken with a contrary sign. 



Substituting these values in the equations (5), and making 

 to abridge 



A' = ( G + A) a 2 + (N+ B) b* + (M+ C) c 2 , 

 B' = (N+ A) a *+(H+B) b* + (L + C) c\ 

 C r = (M+ A) a * + (L + B) 6 2 + (/ + C) c 2 ; 



F'=(N+R)ab', 

 we get = (^' 



+ (B r - pe*) j3 + D'y 

 = E'a +D'j3 + (C" 



These last equations will serve to determine three values of 

 p 2 , and three corresponding ratios of the quantities a, /3, 7; and 

 hence we know the directions of the disturbance by which a 

 plane wave will propagate itself without subdivision, and also 

 the corresponding velocities of propagation. From the form of 

 the equations (6), it is well known, that if we conceive an 

 ellipsoid whose equation is 



1 = A'x z + B'f + C'z* + Wyz + 2E'xz + ZF'xy* ...... (7), 



* If we reflect on the connexion of the operations by which we pass from the 

 function (4) to the equation (7), it will be easy to perceive that the right side of the 

 equation (7) may always be immediately deduced from that portion of the function 



