Theory of a Contractile JEther to Optical Problems. 527 

 The three principal normal velocities then are : — 



\/A/p x , v / A/ / 9 i/7 \/A/p 2 ; 

 while for the nearly transverse waves they are: — 



x/Bji*> y/W* ^Wz- 



The sole condition, therefore, for the disappearance of the 

 normal wave is that A should be extremely small compared 

 with B. This is the same as for an isotropic medium. 



Taking, now, the extreme case in which A vanishes in 

 comparison with B, let us determine the relations between the 

 direction of vibration \, fj,, v, and that of propagation I, m, n. 



If we put A = in (5), we get 



du dv die , 0Q . 



or 



?+?+5-° w 



Now if /, ?n, n are the direction-cosines of a normal to the 

 ellipsoid, 



aV + &y + cV = l, .... (25) 



then the direction-cosines of the line, joining the centre of the 

 ellipsoid to the point of contact of the tangent-plane 



lx + my + nz=l (26) 



are proportional to 



l/a*, m]b\ n/c 2 . 



Hence \, fi, v, or the direction of vibration, lies in a plane 

 normal to that radius vector of the ellipsoid (25) which is 

 drawn to the point of contact of (26). 

 Again, from (9) we have 



Y'-a^-^ilX + mn + nv)^, . . (27) 



and two similar equations. Whence 



■w-V(*+"«-<~>(t-t> X «,, 



and from these, 



^0,W) + ^(c*-«2) + ^V-i3)=0. . (20) 

 Thus, for all values of A and B, the line X, //,, v lies on the 



