DE. J. CASET ON CYCLIDES AND SPHEEO-QUAETICS. 
695 
If five binodal cyclides circumscribed to a cyclide W have one common node , their 
curves of contact with W are five sphero-quartics tying on five spheres having a common 
radical plane. 
283. If : *',ft,y'J'; ci" be two inverse pairs of points with respect to the sphere 
of inversion U of the cyclide W, the binodal cyclides which have these pairs of points 
respectively as nodes, and which circumscribe W, touch W along the sphero-quartics 
in which it is intersected by the polar spheres of (a', ft, y', V), (a", ft', y", b") ; hence the 
points of contact of generating spheres through both pairs lie on the circle of inter- 
section of the polar spheres : but the plane of this circle intersects W in a bicircular 
quartic, and the bicircular quartic and the circle intersect in four points ; hence there 
will be four points of contact, and consequently only two generating spheres can be 
described through two pairs of inverse points. 
This theorem may be otherwise stated. Thus, it is plain that the circle through two 
pairs of inverse points is reciprocal to the circle in which their polar spheres intersect, 
and then we have the theorem, that through any circle can be described two generating 
spheres; their points' of contact are concyclic , and lie on the reciprocal circle. 
284. Since when a sphere of inversion U and a focal quadric F are given the cyclide 
is determined, and if nine points are given the quadric is determined, it hence follows 
that, being given a sphere U and nine spheres which are orthogonal to it, a cyclide can be 
described having U for a sphere of inversion, and the nine spheres as generating spheres. 
285. Since, being given any eight points, three quadrics can be described to touch a 
given plane, we have the theorem, that, being given any eight generating spheres of a 
cyclide, three cyclides can be described through the same pair of inverse points with respect 
to U (see art. 268, 2°), and the cyclides are mutually orthogonal (see art. 119). 
286. If two quadrics intersect in the same eight points, all quadrics passing through 
these eight points have a common curve of intersection. Hence, if two cyclides W, W' 
have eight, generating spheres common, every cyclide having these eight spheres as gene- 
rators will have also as generators all the generators common to W, W'. We shall in 
the next chapter find the equation of the surface formed by all the generators common 
to two cyclides, and also give some of its properties. 
287. Given seven points or tangent planes common to a series of quadrics, then an 
eighth point or tangent plane common to the system is determined. Hence, being given 
seven generating spheres or pairs of inverse points common to a system of cyclides, then 
an eighth generating sphere or pair of inverse points common to the whole system is deter- 
mined. 
288. If a system of cyclides pass through the same eight, pairs of inverse points, their 
polar spheres with respect to a given fair of inverse points have a common radical plane. 
For if P and Q be the polar spheres of a given pair of inverse points with respect to 
W and W', then P-j-/.Q is the polar sphere of the same pair of inverse points with respect 
to W+aW'. 
289. By reciprocating the theorem of the last article we get the theorem : — If a system 
