THE CYCLIDE. 159 



the two rods be then moved always keeping the string tight at the point of 

 intersection R, then R will trace out a conicoid. 



If the rods are in the first and second positions, or in the third and fourth, 

 the conicoid will be an ellipsoid. 



If the rods are in the first and third positions, or in the second and fourth, 

 the conicoid will be an hyperboloid of one sheet. 



If the rods are in the first and fourth positions, or in the second and third, 

 the conicoid will be an hyperboloid of two sheets. 



In the parabolic confocal system, the fourth sheet of the cyclide is a plane, 

 and r t is parallel to the axis of x. Hence if rays parallel to the axis of a 

 paraboloid are reflected by the surface, they will all pass through the two focal 

 parabolas of the system, and the wave surface after reflexion will be a cyclide, 

 and if the rays are twice reflected, they will become again parallel to the axis. 



