496 Dr. J. Casey on Cyclides and Sp hero -Quar tics. [June 15, 



From these equations I have shown that in general a quartic can be 

 generated in five different ways as the envelope of a variable sphere which 

 cuts a given sphere orthogonally, and whose centre moves on a given 

 quadric, which, on account of one of its most important properties, I have 

 named the focal quadric of the cyclide. Every cyclide has, in general, five 

 focal quadrics ; these focal quadrics are confocal ; their focal conies are 

 double, or "nodo-foci" of the cyclide. 



I have shown that the locus of the single or ordinary foci of cyclides 

 are sphero-quartics (curves of intersection of a sphere arid a quadric). In 

 general a cyclide has five focal sphero-quartics. If we call confocal two 

 cyclides having in common one focal sphero-quartic, through any point can 

 be described three cyclides confocal with a given cyclide. These confocals 

 are mutually orthogonal. Other methods of generating cyclides are also 

 given ; tbus three circles in space being given, whose planes are diametral 

 planes of a given sphere, and which are orthogonal to the sphere, a cyclide 

 will be generated by a variable circle in space which rests on these three 

 circles. This method is analogous to that for describing ruled quadrics by 

 the motion of a line. The equation of a cyclide may be interpreted in 

 three different ways : — 1, so as to denote a cyclide ; 2, a sphero-quartic ; 

 3, a tangent cone to the cyclide. Hence it follows that sphero-quartics, 

 both in their modes of generation and in many of their properties, bear a 

 striking analogy to cyclides. Thus the canonical form of the equation of 

 a sphero-quartic is «a 2 + 5^ 2 + cy 2 + ^ 2 =0, where a, /3, y, o are circles on 

 a given sphere U ; the poles of the planes of a, /3, y, I with respect to U 

 are the vertices of the four cones which can be described through the 

 sphero-quartic. The equations of a, /3, y, d are, by incorporating con- 

 stants, connected by an identical relation, a 2 + /3 2 + y 2 + 3 2 =0. By means 

 of this relation, which holds also for bicircular quartics, I have got the 

 equations of the four focal sphero-conics of the sphero-quartic. These 

 sphero-conics are constructed geometrically as the intersections with U of 

 perpendiculars from its centre on the tangent-planes to the four cones 

 which can be drawn through the sphero-quartic. The focal sphero-conics 

 are confocal, their foci being the double or nodo-foci of the sphero-quartic. 



Sphero-quartics may be inverted into bicirculars ; they may also be pro- 

 jected into bicirculars, and that in two ways. First, on either plane of 

 circular section of the quadric, whose intersection with the sphere is the 

 sphero-quartic bylines parallel to the greatest or least axis of the quadric ; 

 second, by elliptic projection — that is, by lines of curvature of confocal 

 quadrics passing through each point of the sphero-quartic. The de- 

 velopable formed by tangent planes to the' sphere U, at every point of the 

 sphero-quartic, possesses many geometrical properties. Thus the cone 

 whose vertex is at the centre of U, and which stands on its cuspidal edge, 

 may be generated by the focal lines of a variable cone osculating one cone 

 of the second degree, and having double contact with another. The 

 cuspidal edge and the nodal lines of the developable may be projected 



