DR. J. CASEY ON CYCLIDES AND SPHERO -QTT ARTI CS . 
G89 
the two planes.” Hence, if a system of cy elides having two common circular generators 
pass through two inverse pairs of points, their director cyclides are inscribed in a hinodal 
Cartesian cy elide, in which is also inscribed the cyclide determined by the two circular 
generators and the circle through the two inverse pairs of points. 
5°. “The director sphere of every ruled quadric passing through the four sides L, M, 
N, P of any skew quadrilateral passes through the circle of intersection of the two 
spheres of which the two diagonals are diameters.” Hence, if L, M, N, P be four circles 
in space,, such that each of the four pairs (LM), (MN), (NP), (PL) is homosplieric, and 
if L, M, N, P be circles on a cyclide W, the director cyclide ofW can be expressed as a 
linear f unction of two Cartesian cyclides, viz. the cyclide which has the spheres (LM.) and 
(NP) as generating spheres, and the middle point between their centres as a triple focus, 
and the cyclide similarly determined by the spheres (MN) and (PL). 
6°. “ When eight planes pass in pairs through any generators of the same ruled 
quadric, the director spheres of all quadrics touching them all have a common circle of 
intersection with that of the original quadric.” Hence, when eight pairs of inverse points 
lie two by two on four circular generators of a given cyclide, the director cyclides of 
all cyclides passing through the eight pairs are such that the director cyclide of the given 
cyclide can be expressed as a linear function of any two of them; in other words, the 
director cyclides of the variable cyclides and the director cyclides of the given cyclides are 
all inscribed in the same hinodal Cartesian cyclide. 
7°. If a system of paraboloids touch the same six planes, their director planes have a 
common point of intersection. Hence, if a system of cubic cyclides pass through the same 
six pairs of inverse points, all their director pairs of points are homosplieric. 
8°. “ If a system of ruled quadrics passing through a common line touch four common 
planes, their director spheres have a common radical axis.” Hence, if a system of cyclides 
having a common circular generator pass through four inverse pairs of common points, their 
director cyclides have two common generating spheres. 
2G8. I shall for the next illustration take the properties of quadrics given in a paper 
of Dr. Salmon’s in the same volume of the Quarterly Journal, “On the Number of 
Surfaces of the Second Degree which satisfy nine conditions.” 
1°. Dr. Salmon proves “ that two quadrics can be described through eight given 
points to touch a given line.” Hence two cyclides having a given sphere of inversion 
can be described having eight given generating spheres to have double contact with a given 
circle. 
2°. 44 Three quadrics can be described through eight given points to touch a given 
plane.” Lienee three cyclides can be described having eight given generating spheres to 
pass through a given pair of inverse points. 
3°. “ Twenty-one quadrics can be described through five given points to touch four 
planes.” Hence twenty-one cyclides can be described having five given generating spheres 
to pass through four given inverse pairs of points. 
4°. In general if N(r, s, t) denote the number of quadrics which can be described to pass 
