DE. J. CASEY OX CYCLIDES AND SPHEEO-QUAETICS. 
603 
199. Let OH intersect AD and B C in P and Q, then P and Q are the limiting points 
of U and F ; and if we denote the radii of U and F by r and E, we have O P . O Q=r 2 , 
and HP . HQ = E 2 ; but since the lines AO, AH are evidently the bisectors at A of the 
supplemental angles made by the tangents, they are at right angles to each other, and 
in like maner the lines OB, BII are at right angles to each other. Hence the points 
Q and O are inverse points with respect to F', and P and O with respect to F". Again, 
it is easy to see that Q and O are inverse points with respect to the imaginary sphere 
XT' whose centre is P, and which cuts U orthogonally, and P and O with respect to U" 
whose centre is Q, and which cuts U orthogonally. Hence the limiting points of XT' and 
F' are the centres of IT" and U; the limiting points of U" and F" are the centres U and IT' ; 
so that the limiting points of any U and its corresponding F are the centres of the two 
remaining U’s or spheres of inversion. 
200. The centres of inversion of a Cartesian cyclide are foci of the surface. 
Demonstration. Let the equations of U and F be 
x 2 -\-y 2j rz 2 =r 2 , and (x af + if fi- A = TP, 
then the perpendicular OT let fall from O, the centre of U on a tangent plane to F, is 
evidently equal to E— a cos 6, where 6 is the angle which the perpendicular makes with 
the axis of x\ and if P, P' be points on OT such that OT 2 — TP 2 =OT 2 — TP' 2 ==r 2 , then 
P, P' are points on the Cartesian cyclide ; and denoting OP by g, Ave have 
2(E— a cos 0')g=r 2 -\-f, 
or 
ZRq=x 2 -\-y 2 -\-z 2 -\-2ax-\-r 2 ; 
that is, 
4E 2 (,r -f -y 2 + z 2 ) = (x 2 -f -y 2 + z 2 -f 2ax + r 2 ) (138) 
Hence x 2 -{- if -\-z 2 = 0 is an imaginary cone circumscribed to the cyclide. Hence the 
centre of U is a focus of the surface. 
201. The equation (138) may evidently be written in the form S 2 =& 3 L, where S is 
a sphere and L a plane, showing that the imaginary circle at infinity is a cuspidal edge 
on the surface. 
The equation of the sphere S is found to be 
x 2 -\-y 2j r z 2 -{-2ax J r r 2 — 2E 2 =0 ; 
and this is concentric with the focal spheres F, F', F" ; but the centre of S is a triple 
focus, as appears from the equation S 2 =£ 3 L. 
Hence the common centre of the three focal spheres F, F', F" is the triple focus of 
the Cartesian cyclide. The Cartesian cyclide has a tangent plane 'which touches it along 
a circle ; the plane is L ; and the circle of contact is the circle of intersection of S and the 
plane L. 
202. Since being given the sphere of inversion IT and the focal sphere F the Carte- 
sian cyclide is determined, we see that a Cartesian cyclide is determined by eight con- 
stants. The same thing appears from the equation S 2 =£> 3 L. From this equation also 
