THE CYCLIDE. 149 



positive and negative ends of the axis of x reversed, we need study the positive 

 values only. 



(1) When r lies between zero and b, the section in the plane of the 

 ellipse consists of two circles, whose centres are the foci, and which intersect at 

 a point of the ellipse. The section in the plane of the hyperbola consists of 

 two circles exterior to each other, whose centres are the vertices of the ellipse. 

 The cyclide itself consists of two lobes exterior to each other, of which the 

 negative one is the largest, and increases with r, while the positive lobe 

 decreases. Each lobe terminates in two conical points, where it meets the other 

 lobe. The cone of contact at these singular points is a right cone, whose axis 

 is the tangent to the ellipse, and whose semi-vertical angle is 



The whole figure resembles two pairs of horns, each pair joined together 

 by their bases, and the two pairs touching at the tips of the horns. Figure I.* 

 represents a cyclide of this kind. The 'continuous curves represent the lines 

 of curvature of both series. The dotted curves represent the ellipse and hyper- 

 bola through which the normals pass, and the dotted straight lines repre- 

 sent the axis of x, and the two straight lines through which the planes of 

 the circles pass. 



(2) When r lies between b and c, the cyclide consists of a single sheet 

 in the form of a ring, the section of which is greatest on the negative side. 

 Figure II. represents a cyclide of this kind. 



(3) When r is greater than c, the cyclide again consists of two sheets, 

 the one within the other, meeting each other in two Conical points which are' 

 situated on the positive branch of the hyperbola. The semi-vertical angle at 

 these points, is 



There is also in all forms of the cyclide, a singular tangent plane which 



touches the cyclide along a circle corresponding to B = - . Figure III. repre- 

 ia 



* [Page 159]. 



