DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
G39 
Cor. Eacli cyclide of the three orthogonal cyclides being a surface of two sheets, hence 
there will be two systems, each consisting of three sheets, and each system will have 
eight points common to all. Hence the three orthogonal cyclides will have sixteen 
points common to all ; these will be eight pairs of inverse points. 
121. The two cyclides 
acd + b\ 3 2 + ay 2 + db 2 -f- ee\ 
a 2 . f3 2 7 2 
„ + h +T+ ,/+ 
have in common their five focal sphero-quartics. 
Demonstration. For eliminate a 2 from these cyclides by means of the identical relation 
* 2 +0 2 + y 2 +& 2 +a 2 , 
and we see, by making k—a in the equation (95), that the cyclides have one focal 
sphero-quartic in common. Hence the proposition is proved. 
122. If tiro cyclides having the same spheres of inversion be reciprocals with respect to 
the square of any of these spheres , then each intersects this sphere in a sphero-quartic of 
foci of the other. 
For it is evident the cyclides 
and 
possess this property. 
aat + b[. 3 2 + cy~ + db 2 
2 /3 2 'y 2 8 2 
fL\i_'L' 4 X+- 
a + b + c + J 
d 
Section II . — Foci of Sphero-quartics. 
123. We have seen that every sphero-quartic can be generated in four different ways 
as the envelope of a variable circle on the surface of a sphere IJ, the centre of the vari- 
able sphere moving along a sphero-conic while it cuts a fixed circle on U orthogonally. 
Now, if one of these sphero-conics beF, and a the corresponding circle, it can be seen, in 
the same way as in art. 104, that each point in which « intersects F is a focus of the 
sphero-quartic. Again, the four cones which stand on the sphero-conics (see equations 
(45), art. 41), and whose common vertex is at the centre of U, are plainly the reciprocals 
of the four cones which can be drawn through the sphero-quartic ; but these latter cones 
have the same planes of circular section, therefore the former system have the same 
focal lines. Hence we have the following theorem, analogous to one in ‘ Bicirculars — 
Every sphero-quartic has sixteen single foci , and these lie four by four on four confocal 
sphero-conics , these sphero-conics being the defer entes or focal conics of the sphero-quartic. 
124. Being given one circle of inversion and the corresponding focal sphero-conic of a 
sphero-quartic, the three remaining focal sphero-conics can be constructed. For circum- 
scribe a spherical quadrilateral to the circle and conic and through the three quartets of 
opposite intersections describe confocals , and we have the thing done. 
125. Let us consider one of the double lines of the developable which circumscribes 
