IN CRYSTALLIZED MEDIA. 137 



Then, by what was before observed, the right side of the equation 

 of the ellipsoid, which gives the directions of disturbance of the three 

 polarized waves and their respective velocities, will be had from <p i} by 

 changing u, v and w into x, y and % ; 



also -j- , — and -=- into a, b and c. 

 ax dy d% 



We shall thus get 



1 = Ax 2 + By* + Czr + 2 Dyz + 2 Ex% + 2 Fxy. 



Provided 



A = (?) a 2 + (/*) c 2 + ( 7 2 ) ¥ + 2 Q3 7 ) Jc + 2 (f )3) «c + 2 (£7) aJ, 



B = (rf) ¥ + (a 2 ) C« + ( 7 2 ) « 2 + 2 (a 7 ) dC + 2 (ijoj &C + 2 (»/ 7 ) a J, 

 C = (£ 2 ) c 2 + (a 2 ) J 2 + (/3 2 ) a 2 + 2 (a/3) a J + 2 (£a) ic + 2 (£/3) ac, 

 D= (^)bc + (a 8 ) £c + (j3 7 ) a 2 + (a/3) «c + (ay) ab 



+ (ar,)b* + (a£)c? + ((Sr,)ab + (y%) UC, 



E = W ac + (/3 s ) ac + (ay) ¥ + (a/3) be + (/3 7 ) ab 

 + 03f) « 2 + OD c 8 + (« D «& + ( 7 D f c, 



F = (f „) «& + ( 7 2 ) ab + (a/3) c 2 + (a 7 ) ic + (/3 7 ) ac 

 + (7?) a 2 + (7,) 6 2 + ( a £) ac + {(ir,) be 



But if the directions of two of the disturbances are rigorously in 

 the front of a wave, a plane parallel to this front passing through the 

 center of the ellipsoid, and whose equation is 



= ax + by + c%, 



must contain two of the semi-axes of the ellipsoid; and therefore a 

 system of chords perpendicular to this plane will be bisected by it; 

 and hence we get 



= (A - C) ac + E (c 2 - a-) + Fbc - Dab, 



= (B- C)bc + J)(c 2 - b*) + Fac-Eab. 

 Vol. VII. Part II. * S 



