158 THE CYCLIDE. 



the curre to the point (-&) of the rod. These strings will determine the point 

 of the rod which rests on any given point of the ellipse. 



Let the rod also rest on the hyperbola, so that either the positive portion 

 of the rod rest* on the positive branch of the hyperbola, or the negative portion 

 of the rod reete on the negative branch. 



Then the point r of the rod lies in the surface of the cyclide whose para- 

 meter ia r, and as the rod is made to slide on the ellipse and hyperbola, the 

 point r will explore the whole surface of the cyclide. 



If we consider any point of space R, the rod will pass through it in four 

 different positions corresponding to the four intersections of the cones whose 

 vertex is R passing through the ellipse and hyperbola. 



The first position, r,, corresponds to the first sheet of the cyclide wlm-li 

 passes through R, If we denote the intersection of the rod with the ellipse by 

 E, and its intersection with the positive and negative branches of the hyperbola 

 by +H and H, then the order of the intersections will be in this case 



E, +H, R. 



The second position, r,, corresponds to the second sheet, and the order of 

 intersections is either 



E, R, +H or -H, E, R. 



The third position, r v corresponds to the third sheet, and the order of the 

 intersections is either 



R, E, +H, or -//, R, E. 



The fourth position, r t , corresponds to the fourth sheet, and the order of the 

 intersections is 



R, -H, E, 

 the letters being always arranged in the order in which r increases. 



The complete system of rays is an example of a linear congruence of the 

 fourth order. 



Now if two rods, each fulfilling the above conditions, intersect at R in any 

 two of these four positions, and if a string of sufficient length be fastened to a 

 sufficiently distant negative point of the first rod, be passed round the point /.'. 

 and l)e fastened to a sufficiently distant negative point of the second rod, ami if 



