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DR. J. CASEY ON CYCLIDES A YD SPHERO-QUARTICS. 
Section III . — Focal Quadric a Paraboloid. 
209. When the focal quadric is a paraboloid, the cyclide becomes a cubic surface 
passing through the imaginary circle at infinity. The varieties of this surface corre- 
spond to those of quartic cyclides, and may be briefly enumerated as follows : — 
1°. When the developable circumscribed to U and F is imaginary, the surface consists 
of two sheets, one of which is a closed surface passing through the centre of U 
and altogether within U. The other sheet, which is its inverse of the first, is an open 
sheet, extending to infinity, which it intersects in a right line. The case considered 
here would occur if F were an elliptic paraboloid, and U in the concavity of it without 
meeting it. 
2°. When the developable is real the surface consists, as in 1°, of two sheets, one of 
which is a closed surface and passes through the centre of U, the other sheet is infinite. 
Each sheet intersects U ; and the part of each sheet internal to FT is the inverse of the 
external part. 
3°. When U intersects F in a single oval, the cyclide consists of one infinite sheet pass- 
ing through the centre of U. 
4°. When U touches F, we have to consider separately the cases where F is an elliptic 
paraboloid and where a hyperbolic paraboloid. 
If F be an elliptic paraboloid, and U touch it on the convex side, the cyclide has a 
conic node, whose tangent cone is imaginary. 
If FT touch F on the concave side, the cyclide has a node whose tangent cone is real ; 
and if FT touch F at an umbilic, the tangent cone to the node is one of revolution : 
lastly, if FT osculate F at an umbilic, the tangent cone becomes a plane ; that is, each sheet 
of the cone opens out into a plane, and the node is a biplanar node whose planes coin- 
cide. In the case where F is a hyperbolic paraboloid, when U touches it we have a 
conic node whose tangent cone is always real, but which becomes a pair of planes if U 
osculate F. 
5°. When FT has double contact with F, the cyclide will be binodal. 
210. In the examination we have given in this and the previous sections of this 
chapter, we have seen that a cyclide of any class can have but three species of node 
(namely, the conic node, the biplanar node, and the uniplanar node), and that these corre- 
spond respectively to ordinary contact of the sphere of inversion and the focal quadric, 
oscular contact, and oscular contact at an unibilic. From this it follows that a cyclide 
can at most have but two real nodes ; and if it has two, they must be conic nodes ; for if 
it had a conic node and a biplanar node, U should touch and osculate F at the same 
time ; that is, U should intersect F in a quartic curve having a double point and a cusp, 
and the plane through the cuspidal tangent and the double point would intersect a 
quartic curve in five points, which it cannot do ; and of course a cyclide cannot, for a like 
reason, have two biplanar nodes. 
By a different mode of reasoning it may be shown that a cubic cyclide cannot have three 
real nodes ; for if it had, the plane through the nodes must intersect the cubic cyclide 
