DE. J. CASEY ON CYCLIDES AND SPHERO-QUARTIC3. 
C35 
Gouknerie, in his memoir “ Sur les Lignes Spheriques,” “ singular focus ” (see 
Liouville’s Journal for 1869). 
106. Let us suppose that we have a system of generating spheres passing through the 
same point P ; from P let there he drawn a tangent cone to the focal quadric F, then 
any edge of this tangent cone meets the quadric F in two consecutive points, and the 
generating spheres whose centres are at these points touch each other at P, consequently 
each edge of the cone is a normal to the cyclide at P as well as being a tangent to F : 
now let us suppose the point P to be on the imaginary circle at infinity, and the normals 
to the cyclide at P will be also tangents to it at P, and we see that the tangent lines to 
the cyclide at the imaginary circle at infinity are also tangent lines to the focal quadric 
F. Hence we have this remarkable theorem : — 
The three focal conics of the focal quadric F of the cyclide are double or nodo-focal curves 
of the cyclide. Compare the corresponding theorem, art. 28 in ‘Bicircular Quartics.’ 
107. Since the nodo-focal curves of the cyclide W are the three focal conics of F, 
they are in like manner the three focal conics of F', F", F w , F lv . Hence the five F’s are 
confocal. 
That is, the five focal quadrics of a cyclide are confocal , and their three focal conics are 
such that each point of any of them is a double focus of the cyclide. 
108. If one of the focal quadrics of a cyclide be a sphere, then the focal conics of this 
sphere reduce to the centre, and the cyclide must consequently have the imaginary circle 
at infinity as a cuspidal edge. Hence all the focal quadrics must be spheres, and these 
spheres must be concentric. 
From the analogy of the corresponding case in ‘ Bicircular Quartics ’ I shall call this 
species of cyclide a Cartesian cyclide. 
The ordinary foci of a Cartesian cyclide being the intersection of the spheres of inver- 
sion with the focal spheres are circles, and it has but one singular or nodo-focus, which 
in this case is a triple focus, namely, the common centre of the focal spheres. 
109. If we take four foci for spheres of reference of a cyclide, since these foci are point 
spheres, the result of substituting the coordinates of any point in one of them will be 
the square of the radius vector to the focus. Hence if we denote the vectors to the four 
foci by §, f, f", the vector equation of a cyclide will be in the form (see art. 29) 
0 , 
n, 
m , 
P > 
r > 
n , 
o, 
l , 
9 . » 
m, 
l, 
0 , 
r , 
t. 
V, 
r , 
0 , 
r, 
9 
§ > 
r 112 
> 
r. 
0. 
110. If the focal quadric of a cyclide be a paraboloid, then the cyclide becomes a cubic 
surface together with the plane at infinity. Hence the focal quadrics of cubic cy elides 
are five confocal paraboloids. 
4 s 
MDCCCLXXI. 
