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* - c 2 ) cos (0o + ft), 



(15) 

 <*) - ( - c 2 ) cos (0 - ft) ; 



observing that s and /, 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 2 - s ' 2 = (a* - c 2 ) sin 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 

 QOy' 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 axes 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 ROy' of the reciprocal ellipsoid is a circle, the right 

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

 plane, which cuts OS in S, is fixed ; but the point of contact T 

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

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

 the point T describes a curve e 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). 



