1 66 On a Dynamical Theory of 



may be transferred to the index-surface, by changing the quan- 

 tities a, b, c into their reciprocals. For example, if the normal 

 to a wave make the angles , 0i with the nodal diameters of 

 the index-surface, the formulae (13) give 



s 2 = i- (a 2 + c 2 ) - i (a 2 - c 2 ) cos (0 + 0i), 



(15) 



s' 2 = -i- (a 2 + c 2 ) - i (a 2 - c 2 ) cos (0 - ft) ; 



observing that s and s', the two normal velocities of propaga- 

 tion, are the reciprocals of the radii of this surface which coin- 

 cide with the wave-normal. Subtracting these expressions, we 



get 



s* _ s " = ( a _ c ) gin sin ft. (16) 



As the position of the tangent plane, at any point T of a 

 biaxal surface, depends on the position of the axes of the section 

 QOi/ made in the generating ellipsoid by a plane perpendicular 

 to OT, it is obvious that when this section is a circle, that is, 

 when the point Tis a node of the surface, the position of the 

 tangent plane is indeterminate, like that of the a^xes of the 

 section ; and it is easy to show that the cone which that plane 

 touches in all its positions is of the second order. Again, when 

 the section R0y f of the reciprocal ellipsoid is a circle, the right 

 line OS is given both in position and length ; and the tangent 

 plane, which cuts O8 in 8, is fixed ; but the point of contact T 

 is not fixed, since the semiaxis OH, to which the right line ST 

 is parallel, may be any radius of the circle ROy'. In this case, 

 the point T describes a curve .in the tangent plane, and this 

 curve is found to be a circle. But both these cases have been 

 fully discussed elsewhere.* 



* See Transactions of the Royal Irish Academy, VOL. xvn. pp. 245, 260 (supra, 

 pp. 25-7, 49-51). 



