156 THE CYCLIDE. 



The normal to the ellipsoid through R will be the real axis of the cone 

 which pane* through the ellipse, and will bisect the angle between r, and r,, 

 and also tliat between r, and r,. If the ellipsoid were reflective, a ray incident 

 in the direction r, would be reflected in the direction of r, reversed ; hence, 

 by the wave theory, r, + r, is constant for the ellipsoid. At the point of tin- 

 ellipsoid (p constant) where it is cut by the axis of z, 



r, = x + c, 



So that the equation of the ellipsoid (r = constant) may be expressed in terms 

 of r, and r, thus : 



2p ...................................... (30). 



The normal to the ellipsoid also bisects the angle between r, and r it whence we 

 deduce another form of the equation of the same ellipsoid 



r, + r t =-2p ..................................... (31). 



Hence, the general relation among the values of r, 



.............. . .................... (32). 



The normal to the hyperboloid of one sheet (/* = constant) bisects the angle 

 between r, and r,, and also that between r, and r tt whence we obtain the 

 equations 



............................ (33). 



The normal to the hyperboloid of two sheets (v = constant) bisects the angles 

 between r, and r,, and between r, and r t , whence 



(34). 



These are the equations to the conicoids in terms of the four rays of the pencil. 

 The equations to the four cyclides in terms of elliptic co-ordinates are easily 

 deduced from them 



(35). 



