658 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
^ass through a common line. This common line is an edge of the cone A 2 +B' J -j-C' 3 =0 
passing through the six cuspidal edges. 
183. Let us consider an edge of the cone (129). It pierces U in two points; these 
are the limiting points of two inverse osculating circles of the sphero-quartic WIT. 
The equation of the locus of these limiting points is easily found ; for the tangential 
equation of the nodal line of % is the equation got by substituting X, v in place of 
x, y, z in the equation of the cone. Hence, if a , [3, 7 be the three circles of inversion of 
WU, the poles of whose planes are at the three remaining vertices of the tetrahedron, 
the equation of the required locus will be got by substituting in the equation (129) 
a, (3, 7 for x, y, z, and therefore it will be 
(J-c){^«)«¥+(c-ff)(#F}H(«-^){#)7 ! }-0, . . . (131) 
a curve which has twenty four cusps. 
184. If a/, y\ z', w' be the coordinates of any point in the sphero-quartic WU, then it 
follows from equation (121), combined with art. 36, that the equation of the osculating 
circle of WU at the point x f z' w' is 
4(a)^( a )+y(iW)(P)+y(cy’(y)+W^l=0, 
where a, /3, 7 , * are the circles of reference when the sphero-quartic is given by its cano- 
nical form. 
CHAPTER X. 
Classification of Cy elides. 
185. Following the analogy of the method given in my memoir on ‘Bicircular 
Quartics,’ I shall take as the basis of classification the species of the focal quadric. 
The principal varieties of quadrics are : — 1°. An ellipsoid or hyperboloid. 2°. A sphere. 
3°. A paraboloid. We shall find the cyclides corresponding to these varieties to have 
fundamental distinctions. We shall therefore devote a section to each. 
Section I. — Focal Quadric an Ellipsoid or Hyperboloid. 
186. Figure of cyclide. Let us denote the sphere of inversion by U, and the focal 
quadric by F. 1°. When the developable circumscribing U and F is imaginary, as for 
instance when F is an ellipsoid and U entirely within it, the cyclide evidently consists 
of two distinct sheets, which are inverse to each other with respect to U. One sheet is 
internal to IT, and the other external ; each sheet is a closed surface. 
2°. When the developable is real, and when U does not intersect F, or else when it 
does intersect it in a sphero-quartic consisting of two distinct ovals, the cyclide W is 
made up of two closed surfaces, each of which is an anallagmatic, and divided by U into 
two parts. The points where W cuts U are the points of contact of the common tangent 
developable circumscribed to U and F. 
3°. When the developable is real and the sphero-quartic of intersection of U and F 
