636 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
111. If we are given a sphere of inversion and the corresponding focal quadric of a 
cyclide, we can construct the remaining focal quadrics. For by art. 34 let a be the sphere 
of inversion and F the focal quadric, and circumscribing a developable 2 to a and F, 
if the double lines of S, which are conics, be C, C', C", C'", then through C, C', C", C" 
let confocals F, F", F", W" to F be described, and these will be the other focal quadrics 
of the cyclide. 
Now if a touches F the developable 2) will have but three double lines, C, C', C", and 
hence in this case there will be only four focal quadrics. If « touches F the cyclide will 
have the point of contact for a node, and moreover the cyclide will be the inverse of 
a central quadric. Hence it follows that the cyclide which results from inverting a central 
quadric has but four focal quadrics , and that three of these confocal quadrics pass through 
the node (see art. 78). 
112. If the sphere of inversion be an osculating sphere of the focal quadric F, the 
developable 2 will have but two nodal lines, C, C, and therefore there will be but two 
additional focal quadrics, F', F". Hence in all there will be but three focal quadrics. 
This is the species of cyclide which results from inverting a non-central quadric (see 
art. 79). 
113. If the sphere of inversion has double contact with F the cyclide will be binodal; 
there will be, besides the focal quadric F, the two confocals to F, which can be drawn 
through the two points where a, touches F. A cyclide of this form being given by the 
equation (a, b, c,f, g, hffi, y, c)) 2 , and the locus of the centre of the generating sphere 
being a conic, it must be a focal conic of the three confocal quadrics which the cyclide must 
have , that is, every point of this conic must be a double focus of the cyclide, and moreover 
the four points in which it intersects the corresponding sphere of inversion must be single 
foci. 
When the sphere of inversion a has double contact with F, the curve of intersection 
of a and F breaks up into two circles ; these circles are the inverses of the two focal 
lines of the cone, of which this species of cyclide is the inverse. 
114. Since three of the spheres of inversion of a cyclide which has only four spheres 
of inversion, and which is consequently the inverse of a central quadric, are the inverses 
of the three principal planes of the quadric, and since the inverse of a focus is a focus, 
it follows that in this case the inverses of the three focal conics of the quadric inverted 
will be the focal sphero-quartics of the- cyclide. In this case also we have the following 
theorem, which is an extension of one given in my ‘ Bircirculars,’ art. 55 : — 
If we invert a quadric Q from any point P, the principal planes of the focal quadrics 
of the resulting cyclide are parallel to the tangent planes at P drawn to three confocals 
of Q passing through P. 
115. In like manner two of the spheres of inversion of a cyclide which has only three 
spheres of inversion, and which is the inverse of a non-central quadric, are the inverses 
of the two planes of symmetry of the quadric; and since the focal conics of a para- 
boloid are either an ellipse and parabola or hyperbola and parabola, we see that one of 
