THE CYCLIDE. 



The third sheet, corresponding to r a , has also three different forms. It 

 may be either the positive lobe of (l), the ring cyclide ( 2), or the outer 

 sheet of ( 3). In the first case it has a conical point on the ellipse where 

 it meets the second sheet. In the second case it has no conical point, and 

 in the third it meets the fourth sheet in a conical point on the negative 

 hyperbola. 



The fourth sheet is the interior spindle of the cyclide ( 3), and always 

 meets the third sheet at a conical point on the negative hyperbola. 



Parabolic Cyclides. 



When the values of b, c, r, and x are each increased by the same quan- 

 tity, and if this quantity is indefinitely increased, the two conies become in 

 the limit two parabolas in perpendicular planes, the focus of one being the vertex 

 of the other, and the cyclide becomes what we may call the parabolic cyclide. 



When r lies between b and c, the cyclide consists of one infinite sheet, 

 lying entirely between the planes a; = 26 r and x = 2c r. The portions of space 

 on the positive and negative side of the sheet are linked together as the earth 

 and the air are linked together by a bridge, the earth, of which the bridge 

 forms part, embracing the air from below, and the air embracing the bridge 

 from above. In fact the earth and bridge form a ring of which one side is 

 much larger than the other. 



A parabolic ring cyclide in which 2r = b + c, is represented in Figure IV. 



When r does not lie between b and c, the cyclide consists of a lobe with 

 two conical points, and an infinite sheet with two conical points meeting those 

 of the lobe. 



Surfaces of Revolution. 



When 6 = 0, the cyclide is the surface formed by the revolution of a circle 

 of radius r about a line in its own plane distant c from the centre. If r is 

 less than c, the form is that of an anchor ring. If r is greater than c, the 

 surface consists of an outer and an inner sheet, meeting in two conical points. 

 When b = c, the cyclide resolves itself into two spheres, which touch externally 

 if ) is less than b, and internally if r is greater than b. When b = c = 0, the 

 two spheres become one. 



