IN CRYSTALLIZED MEDIA. 135 



Fresnel supposes that the wave-velocity depends on the direction 

 of the disturbance only, and is independent of the position of the 

 wave's front. Instead of assuming this to be generally true, let us 

 merely suppose it holds good for these three principal waves. Then 

 we shall have 



N+ A = C + L, M + A = B + L and B + N = C+M; 



or, we may write 



A-L = B-M=C-N= V . (Suppose.) 



Thus our equation (8) becomes, since a 2 + b 2 + c 2 = 1, 



1 = M (ax + by + ess) 2 + v (x 2 + y 2 + as 2 ) 

 + (La> + Mb 2 + iVV) (of + y 2 + s 2 ) 

 + L (cy — b%) 2 + M (a% - ex) 2 + JV (bx — ayf. 



But the two last lines of this formula easily reduce to 



(M +N)x 2 + (N+ L)f + (L + M)% 2 

 + L \a 2 x* - (by + c%) 2 } + M\b 2 y 2 - {ax + ess) 2 } 

 + iV {<?% 2 — (ax + by) 2 }, 



and hence our last equation becomes 



1 = (v + M + N) x 2 + + N + L) f + (v + L + M) «» 



+ m (ax + by + ess) 2 



+ L \a 2 x 2 - (by + ess) 2 } 



(11.) 

 + M{b 2 y 2 - (ax + ess) 8 } 



+ N{c 2 z 2 - (ax + by) 2 }. 



In consequence of the condition which was satisfied in forming the 

 equation (8), it is evident that two of its semi-axes are in a plane 

 parallel to the wave's front, and of which the equation is 



= ax + by + ess, (12.) 



