DE. J. CASEY ON CYCLIDES AND SPHERO-QUAETICS. 
659 
consists of one oval, W consists of one closed surface which is divided into two parts 
by U. 
4°. When U touches F, the point of contact will be a nodal point on the cyclide, the 
cone of contact with the cyclide being’ real or imaginary according as U touches F on the 
concave or convex side of F (see art. 76). 
Cor. If a cyclide has either a real or imaginary conic node (contracted by Professor 
Cayley into cnic-node), it arises from a real double point or a conjugate point on one of 
its focal sphero-quartics. 
5°. When U has stationary contact with F, the point of osculation will be a biplanar 
node on the cyclide. In this case the cyclide will be the inverse of a non-central quadric 
(see art. 76). 
6°. When U has double contact with F, the cyclide will be binodal. 
7°. When U is inscribed in F (that is, when U touches F along a circle), the cyclide 
will break up into two spheres. 
187. Double Tangent Cones. — Let us consider a cyclide whose focal quadric is F ; then, 
taking the limiting points P, P' of IT and any tangent plane to F, the generating sphere 
through P, P' will become a plane if its centre be at infinity, and the locus of the points 
P, P' will evidently be a sphero-quartic, which is given as the intersection of a sphere 
concentric with F, and a cone whose edges are perpendicular to the tangent planes of 
the asymptotic cone of F, the vertex of the cone being the centre of IT ; this cone will 
be a double tangent cone. Hence we have the following theorem : — Every cyclide has 
as many double tangent cones as it has focal quadrics. 
188. The lines of intersection of a cyclide with its spheres of inversion are lines of 
curvature on the cyclide . 
For let us consider any point on the cuspidal edge of the developable which circum- 
scribes IT along WU ; then that point is the centre of an osculating sphere of W (see 
art. 169). Hence WU is a line of curvature on W. 
Cor. 1. The cuspidal edge of 2 is a geodesic on the surface of centres of W. 
Cor. 2. The sphero-quartic reciprocal to W with respect to U 2 (that is, to « 2 -f j3 2 -j- f -j- b' 2 ) 
is such that the focal sphero-quartic of W lying on the sphere U is a line of curvature on it. 
189. Binodal Cy elides of a Cyclide. — We have seen (art. 33) that the cyclide W may 
be written in five different ways, (I.), (II.), (III.), (IV.), (Y.). Now taking the first (I.), 
its equation is 
{a — b)(3' 2J r ( a—c)y 2 -j- [a — d)b 2J r ( a — c)s 9 ~0, 
and the square of the corresponding sphere of inversion is [3 2 J r f -fi o 2 -f g 2 ; and eliminating 
in succession each of the four letters /3 2 , y\ h 2 , s 2 , we get four binodal cyclides, each 
touching W along the line of curvature WU. Hence every cyclide has in general four 
times as many binodals inscribed in it as it has sph eres of inversion. 
199. The imaginary circle at infinity is a fiecnodal curve on the surface of centres of 
a cyclide. This proposition is an extension of art. 52 in ‘ Bicircular Quartics.’ It is 
proved as follows : — It is evident that the normal at any point of the imaginary circle at 
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