DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
621 
This species of node is called by Professor Cayley a binode (see his “ Cubic Surfaces ” 
in the Philosophical Transactions for 18G9, p. 231). 
3°. If 5=0 in the equation of a non-central surface the two planes become coincident, 
that is, the quadric cone becomes a coincident plane pair. Professor Cayley calls 
this species of node a unode. Hence the inverse of a parabolic cylinder is a unodal 
cyclide. 
77. If we invert a binodal cyclide, aod -\-bf~-\-cy 1 , from one of the points common 
to the spheres of reference a , (3, y, the spheres a, (3, y will be inverted into three planes, 
and therefore the inverse of a binodal cyclide from one of the nodes will be a cone of the 
second degree. Conversely, the inverse of a cone of the second degree will be a binodal 
cyclide. 
This conclusion may also be inferred otherwise; for as in 1°, art. 76, the inverse of 
the cone 
( r -«) 2 ( y - 5) 2 C - c ) 2 
L M ' N u 
is the cyclide 
fax by 
L+M+n) (^ 2 +r + s2 ) 2 — L +M + n) (^ 2 +/+ 22 )+^ 4 ( L+M+n) =0; 
and it is evident from the form of this equation that the origin is a conic node. Again, 
if we transfer the origin to the point «, b, c, it will be seen that the new origin is a conic 
node. Hence the inverse of a cone has two conic nodes. 
Cor. If we invert a quadric from a point on the quadric we get a cubic cyclide. 
78. If one of the spheres of inversion touch one of the focal quadrics of the cyclide at 
one point, the focal quadric and the sphere of inversion can be expressed by the two 
tangential equations 
ah 2 -\-b(jh -j -cr +2 nvg =0,| 
a'X 2 + Vg? -f c'v 2 + 2 n'vg = 0. j 
See ''Cayley “ On Developable Surfaces of the Second Order,” Cambridge and Dublin 
Mathematical Journal, vol. v. p. 51. 
Hence it follows that the equation of the cyclide and the square of its sphere of 
inversion U are given by the equations 
a a 2 -J-5/3 2 -\-cy 2 +2 nyb =0 5 ) 
a! a 2 + Vf + c'y-+2n'yl = 0,/ 
where a, f3 are spheres of inversion of the cyclide, y a point sphere, which is a centre of 
inversion of the cyclide in the sense that it is the centre of a circle which inverts the 
cyclide into a quadric. Hence in this case there are four centres of inversion, namely, 
the centre of U and the centres of a, (3, and the point sphere y. 
The spheres of inversion are, 1°, the sphere which inverts the quadric into a cyclide ; 
2°, the inverses of the principal planes of the quadric. 
The quadric must be either an ellipsoid or a hyperboloid ; and the cyclide, which is its 
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