594 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
which it is necessary to examine, as this species of cyclide will occupy much of our space 
in the present memoir. 
Since all the tangent planes of a cone pass through the same point, and since every 
tangent plane determines two points on the cyclide, it is plain that all the points lie on 
the surface of a sphere whose centre is at the vertex of the cone. 
Again, since the cone is a ruled surface, each edge of it will determine, as in art. 19, 
a circle which will be a generating circle of the cyclide ; but the circle will not in this 
case lie altogether in the envelope as in art. 19, because in art. 19 the points of contact 
of any line on the quadric are distinct for all the planes passing through it, whereas in 
the cone only one tangent plane, properly so called, can be drawn through any edge of it. 
But although the circles which answer to each edge of the cone do not lie altogether in 
the cyclide, yet the envelope of these circles is the cyclide, which in this case is evidently 
a twisted curve, which, as will be shown, is of the fourth degree. On this account 1 have 
called this species of cyclide, for the sake of distinction, a sphero-quartic. 
22. Since the planes of the generating circles in the last article are perpendicular to 
the edges of the focal cone, the envelope of these planes is another cone ; and as each 
plane passes through the centre of the sphere U, the vertex of the second cone is at the 
centre of U. Hence the sphero-quartic is the intersection of a sphere and a cone. 
Hence we have the following theorem : — When a cyclide reduces to a sphero-quartic , it 
is the intersection of a sphere and a quadric . 
23. If we denote the sphere on which we have proved the sphero-quartic lies by O, 
then the generating circles are circles on O ; and as O evidently cuts U orthogonally, the 
circle of intersection of U and O, which we denote by J, will be orthogonal to all the 
generating circles, and the focal cone pierces O in a sphero-conic. Hence we have the 
following method of generating sphero-quartics : — 
Being given a circle J on a sphere, and a sphero-conic on the same sphere. A sphero- 
quartic is the envelope of a variable circle whose centre moves along the sphero-conic , and 
which cuts the circle J orthogonally. 
24. From the last article we infer this other method of generating sphero-quartics. 
Let F be a sphero-conic on a sphere Q, U a point on the surface of O ; from U draw 
an arc UT perpendicular to any great circle tangential to F, and take two points, P, P', 
such that 
cos UT : cos TP=cos UT : cos TP '=!c, 
where Jc is a constant. 
The locus of the points P, P' is a sphero-quartic. 
Cor. If k=l the point P coincides with U, and the point will in this case be a double 
point in the sphero-quartic, and the sphero-quartic itself will be the inverse of a plane 
conic from a point outside the plane of the conic. In fact if the sphere O be inverted 
into a plane from the point U, it is easy to see that the point P' will be inverted into a 
point whose locus is a conic. 
