164 On a Dynamical Theory of 



the surface of the ellipsoid. The equation of the cone B is 

 found from that of A, by changing the coefficients of the squares 

 of the variables into their reciprocals, and is therefore 





which, of course, is also the equation of the wave-surface, if r be 

 supposed to be the radius drawn from to the point whose 

 co-ordinates are x, y, z. Combining this equation with that of 

 the sphere, we have 



0^ ?/ 3 2 



^ + * + ~~ = *' 



which represents a hyperboloid passing through the common 

 intersection of the sphere, the cone , and the wave-surface. 



Since the differences between the coefficients of the squares 

 of the variables in the equation (10) are the same as the corre- 

 sponding differences in the equation of the ellipsoid, the cone A 

 has its planes of circular section coincident with those of the 

 ellipsoid. The cone J5, being reciprocal to A, has therefore its 

 focal lines perpendicular to the circular sections of the ellipsoid. 

 These focal lines are consequently the nodal diameters* of the 

 wave-surface, that is, the diameters which pass through the 

 points where the two sheets of that surface intersect each 

 other. 



If the direction of OT cut the other sheet of the wave- 

 surface in T', and if two radii of constant lengths, equal is OT 

 and OT' respectively, revolve within the surface, the cones B 

 and B' described by these radii will intersect each other at right 

 angles, since they have the same focal lines. And supposing 

 the axis of y to be the mean axis of the ellipsoid, so that the 

 nodal diameters lie in the plane of trz, the axis of x will lie 

 within one of the cones, as , and the axis of z within the other 

 cone B'. Now the angle contained by the two sides of either 



* See Transactions of the Royal Irish Academy, VOL. xvn. p. 247 (supra, p. 29). 



