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DK. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
spheres of inversion, (a) for instance, then the spheres orthogonal to U passing respect- 
ively through two triads of consecutive pairs of points at P, P' will be osculating spheres 
of W, and their circles of intersection with U will be osculating circles of WU. The 
radical plane of the inverse pairs of osculating spheres will be a diametral plane of U, 
and will intersect the face of the tetrahedron in a line which will be a tangent line to 
the curve (115). Hence we have the following theorem: — The envelope of the radical 
plane of a pair of inverse osculating spheres of a sphero-guartic is a cone of the fourth 
degree possessing the following properties : — 
1°. It has three double edges passing through three vertices of the tetrahedron. 
2°. It has six stationary tangent planes. 
3°. If through ang edge four tangent planes be drawn , their edges of contact are corn- 
planar. 
4°. The (inharmonic ratio of the four edges of contact is constant. 
5°. The envelope of the plane through the four edges of contact is a cone of the second 
degree touching the six stationary tangent planes. 
170. Let K be one of the vertices of the tetrahedron, and S one of the osculating 
circles of WU. I say the cone V, whose vertex is Iv and which stands on S, will have 
double contact with the cone whose vertex is K and which circumscribes U. 
Demonstration. The cone which circumscribes U along S, and the cone whose vertex 
is at K and which circumscribes U, have plainly two common tangent planes ; and these 
will evidently be tangent planes to V also. Hence the proposition is proved. 
171. The cone V osculates the cone through WU having the same vertex as V. 
This is evident, since S passes through three consecutive points of WU. The planes 
of circular section of Y are parallel to the plane of S, and to the plane of the inverse 
of S. 
172. If we form the reciprocal of the cone Y with respect to U, its vertex will be at 
the centre of U, its intersection with U will be a sphero-conic having double contact 
with a circle of inversion (see art. 170), (2°) osculating the corresponding focal sphero-conic 
(art. 171); 3°, the focal lines will pass through two points on the cuspidal edge of the 
developable A circumscribed along WU (art. 171). Hence we may enunciate the 
following theorem : — If J and F denote the two cones whose vertices are at the centre of 
U, and which stand respectively on cl circle of inversion and on a focal sphero-conic of 
the sphero-guarticUVXh , the cone standing on the cuspidal edge of A is generated by the 
focal lines of a variable cone which has double contact with J and which osculates F. 
173. The theorem of the last article has an analogue in the theory of bicircular 
quartics. This may be inferred from the one for sphero-quartics ; but the following is a 
direct proof. 
First Ave have to find the locus of the centre of a variable circle which touches one 
circle and which is orthogonal to another. 
Let the variable circle be 
x 2 +y 2 +‘2gx+2fy + c= 0, 
