308 ON THE PROPAGATION OF LIGHT 



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

 equation of the ellipsoid, which gives the directions of dis- 

 turbance of the three polarized waves and their respective 

 velocities, will be had from < 2 by changing u, v, and w into 

 a?, y } and z ; also 



-=- , -7- , and -j- into a, b, and c. 

 dx dy dz 



We shall thus get 



1 = Ax* + % 2 + Cz* + ZDyz + 2Exz + ZFxy. 

 Provided 

 4 = C? 2 ) ' + (') c 2 + ( 7 2 ) 5 s + 2 O&y) 6c + 2 () ac + 2 (fy) ok, 



B= (rf) W + (a 2 ) c 2 + ( 7 2 ) a a + 2 (07) ac + 2 (973) 5c + 2 (177) o&, 



> 



^= (D c 2 + (a 2 ) 5 2 + G8 2 ) a 2 + 2 (a/9) a& + 2 (fa) be + 2 (fiS) ac, 



D= fef) 5c + (a 2 ) fo + (^7) a 2 + (a) ac + (a 7 ) a6 



+ (<*!) 6 2 + (af) c 2 + 0&?) a5 + ( 7 f) ac, 

 ^= (f f ) ac + OS 8 ) ac + (07) b 3 + (a/3) 6c + (7) a& 



+ (7 s ) o* + (a/3) c 2 + (a 7 ) be + (^7) ac 



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 centre of the ellipsoid, and whose equation is 



= ax + by + cz, 



must contain two of the semi-axes of this ellipsoid; and there- 

 fore a system of chords perpendicular to the plane will be 

 bisected by it; and hence we get 



0= (A - C) ac + E^-a^+Fbc-Dab, 

 = (B - C) be + D (c 2 - 6 2 ) + .Fac - Eab. 



