662 Dll. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
Section II . — Focal Quadric a Sphere. 
197. When the focal quadric is a sphere, the cyclide has the imaginary circle at 
infinity as a cuspidal edge ; on this account we shall call the surface a Cartesian cyclide. 
Figure of the Surface. 
1°. When U is external to F, W consists of two distinct sheets, each intersecting U 
in a circle. Each sheet is a closed surface. 
2°. When U is internal to F, W consists of one sheet internal to U, and another sheet 
the inverse of the former, and therefore external to U. Each sheet is a closed surface. 
3°. When U intersects F, W consists of one sheet ; this intersects U in one real circle 
and another imaginary circle. The sheet is a closed surface. 
4°. When U touches F internally, W has a conic node, the tangent cone to W at the 
node being a real cone of revolution. 
5°. When U touches externally, W has a conic node at which the tangent cone is 
imaginary. 
6°. When U reduces to a point, W is the pedal of a sphere, and is therefore the inverse 
of a quadric of revolution from the focus. 
198. In the annexed diagram, which is supposed to be a plane section through the 
centres of U and F, let B, C and A, D be the opposite pairs of the intersections of the 
tangent cones circumscribed about U and F made by the plane of the section, then the 
spheres F, F" concentric with F, and passing respectively through B, C and A, D, are 
Fig. 5. 
focal spheres (see art. 34) of the Cartesian cyclide ; for the developable circumscribed 
about U and F reduces in this case to two cones of revolution, and the nodal lines of the 
geometric system composed of the two cones are two circles which are intersected by the 
plane of the section in B, C, A, D and the vertices of the cones. Hence the line O H 
passing through the two vertices must be regarded as a limiting case of an hyperboloid 
confocal with the spheres F, F', F" (see art. 106). Hence the focal quadrics of a Carte- 
sian cyclide are three concentric spheres and a straight line through their centre. 
