688 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
in which these lines intersect the cyclide , the envelope of the sphere through P, P', Q, Q', 
R, B/ is a Cartesian cyclide , which by analogy I shall call the director cyclide of the given 
cyclide. 
1°. “ If a system of quadrics touch a common system of eight planes, their director 
spheres have a common radical plane.” lienee, if a system of cy elides pass through a 
common system of eight inverse pairs of points, their director Cartesian cyclides are 
inscribed in a common binodal Cartesian cyclide. 
2°. “ If a system of quadrics touch a common system of seven planes, their director 
spheres have a common radical axis.” Hence, if a system of cyclides pass through a 
common system of seven inverse pairs of points, their director Cartesians have two generating 
spheres common to all. 
3°. “ If a system of quadrics touch a common system of six planes, their director 
spheres have a common radical centre.” Hence, if a system of cyclides pass through a 
common system of six inverse pairs of points, their director Cartesian cyclides are such 
that any of them can he expressed as a linear function of four others , because their 
director spheres in the property of the quadrics having- a common radical centre are 
coorthogonal, and any of them can be expressed as a linear function of four others. 
Cor. The property in 1° may be expressed thus : — any director Cartesian can be 
expressed linearly in terms of two others ; and the property in 2°, any director Cartesian 
can be expressed linearly in terms of three others. 
267. When a quadric becomes a paraboloid, the director sphere becomes a director 
plane. Hence if three lines mutually perpendicular be drawn through a centre of 
inversion of a cubic cyclide intersecting it in three pairs of inverse points P, P', Q, Q!, 
P, B/, the sphere determined by these three pairs of inverse points passes through a fixed 
pair of points. I shall call these points the director points of the cubic cyclide. 
1°. “ If a system of quadrics touch the same eight planes, the common radical plane 
of their director spheres is the director plane of the paraboloid which touches the planes.” 
Hence, if a system of cyclides pass through a common system of eight inverse pairs of 
points, the nodes of the binodal Cartesian cyclide which is circumscribed to their director 
Cartesian cyclides are the director points of the cubic cyclide which passes through the 
system of eight pairs of inverse points. 
2°. “ If a system of paraboloids touch the same seven planes, their director planes have 
a common line of intersection.” Hence, if a system of cubic cyclides pass through a common 
system of seven pairs of inverse points, their director points are coney die. 
3°. “ If a system of quadrics having a common rectilinear generator touch five planes, 
their director spheres have a common radical plane.” Hence, if a system of cyclides 
having a common circular generator pass through five inverse pairs of points, their director 
cyclides are inscribed in the same binodal Cartesian cyclide. 
4°. “ If a system of ruled quadrics have two common rectilinear generators and touch 
two common planes, their director spheres have a common circle of intersection with 
that of the ruled quadric passing through the two lines and through the intersection of 
