DR. J. CASEY ON CYCLIDES AND SPHERO-QDARTICS. 
G67 
in three right lines, and then, by inversion, from any point in the plane we should have 
the absurdity of a quartic cyclide being- intersected by a plane in three circles. 
211. Parallel Tangent Planes. — Let us consider a cubic cyclide whose sphere of inver- 
sion is U and focal paraboloid F. If P, P' be the limiting points of U and the tangent 
plane to F at infinity, then of the two points P, P r one must be at the centre of U and 
the other at infinity ; and it is plain that the generating sphere which touches the cyclide 
at P, P' must break up into two planes, namely the tangent planes to the cyclide at 
P, P'. The tangent plane at the centre of U must evidently be parallel to the principal 
plane of the paraboloid which does not intersect it in a parabola ; and since the cyclide 
has five centres of inversion, we see that every cubic cyclide has jive parallel tangent 
planes , and these are the jive tangent planes which can be drawn to a cubic cyclide jrom 
the line at injinity on the cyclide. lienee we infer the following theorem : — The jive 
tangent planes to a cubic cyclide from the line at infinity on the cyclide have the five 
centres of inversion as points of contact. 
212. The property of the last article may be shown otherwise. Thus, consider any 
quartic cyclide; then at any point Q five generating spheres touch, namely one belonging to 
each of the five systems of generating spheres. Now, since a generating sphere intersects 
a cyclide in two circles, if we invert the quartic cyclide from the point Q we get a 
system of five parallel planes, each intersecting the cubic cyclide into which the quartic 
inverts in two lines, and therefore having the points of intersection of these lines as 
points of contact with the cyclide. 
213. The section of a cubic cyclide made by a plane passing through any line on the 
cyclide except the line at infinity must consist of the line and a circle ; for it must con- 
sist of a line and a conic, and by inverting from any point the line and conic must 
invert into a bicircular quartic ; hence the conic must be a circle. 
This reasoning will not apply in the case of a section made by a plane through the 
line at infinity ; for when we invert the line at infinity it becomes a point, which accounts 
for the double point which results when a conic is inverted ; so that when we say the 
inverse of a plane conic is a bicircular quartic, this includes the inverse of the line at 
infinity together with that of the conic. 
214. Since the five centres of inversion of a cubic cyclide form a pentahedron, such 
that, taking any four of them forming a tetrahedron, the perpendiculars of that tetra- 
hedron are concurrent and intersect in the fifth point, w 7 e see without difficulty that 
the feet of the perpendiculars of the pentahedron are points on the cubic cyclide, and 
we may easily infer the following theorem : — 
Being given eight liomospheric points {say, eight points on the sphere U), three cubic 
cyclides can be described having these eight points as foci. These cyclides intersect two 
by two orthogonally , they have the same five centres of inversion, and each passes through 
the feet of the perpendiculars of the pentahedron. 
215. In order to find the equation of the tangent cone to a cubic cyclide from any 
given point, let the given point be taken as the origin of Cartesian coordinates, and 
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