622 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUABTICS. 
inverse, will have the point of contact of the two surfaces (79) for a node. It is plain 
this cyclide is the pedal of a quadric. Fresnel’s surface of elasticity is an example. 
The generating spheres will be, 1°, the inverses of the three systems of spheres whose 
centres are in the principal planes of the quadric, and which have double contact with 
it ; 2 °, the inverses of the tangent planes of the quadric. 
There will be four focal quadrics, namely, the loci of the four systems of generating 
spheres. 
79. If one of the spheres of inversion osculate the focal quadric, the tangential 
equation of the focal quadric and sphere of inversion can be expressed by the system 
l (X 2 — 2[Lv) + 111 ((J 2 — 2ve) = 0, ) 
l'(X 2 — — 2^)=0 J 
(see Cayley, supra). 
Hence the cyclide and the square of the sphere U may be expressed by the system of 
equations 
?(«»-2ft0+™'(P*-V)=Oj 1 
where (3 is a sphere of inversion, 7 a point sphere, which is a centre of inversion in the 
sense that it is the centre of a sphere which inverts the cyclide into a paraboloid. 
Hence in this case there are only three centres of inversion, namely, the centre of U, 
the centre of /3, and the point sphere y=0. 
The three spheres of inversion are U, /3, and the sphere whose centre is 7 , which 
inverts the paraboloid into a cyclide. The spheres IT and /3 are the inverses of the two 
planes of reflection of the paraboloid. 
The generating spheres will be, 1°, the inverses of the two systems of spheres whose 
centres are in the two planes of reflection, and which have double contact with the 
paraboloid; 2 °, the inverses of the tangent planes of the paraboloid. 
There will be three focal quadrics, namely, the loci of the centres of the three systems 
of generating spheres. 
80. The binodal cyclide aa 2J r b(3 2 -\-cy 2 =0 will have, besides the centres of the spheres 
u, (3, 7 , which are centres of inversion, an infinite number of centres of inversion lying 
on the line joining the two nodes. 
This is the species of cyclide that is generated when the sphere of inversion U has 
double contact with the focal quadric F ; each point of contact will plainly be a node of 
the cyclide ; and since the surface is the inverse of a cone, there will be, as in the case of 
cyclides which are the inverses of hyperboloids, four systems of generating spheres*. 
* [In writing this memoir I omitted trinodal and quadrinodal cyclides. My attention was directed to this 
omission by Professor Cayley. This omission was the less excusable, inasmuch as quadrinodal cyclides were 
the only surfaces to which the name cyclide was applied, until M. Darboux extended the term (see ‘ Comptes 
Itendus,’ June 7 , 1869). The existence of trinodal cyclides was first proved by Professor Cayley in the Quar- 
