DK. J. CASEY ON CYCL1DES AND SPHEKO-QUAKTICS. 
G41 
130. Let us denote the planes of circular section of the cones through WU by P, P', 
then P passes through two of the four circular points at infinity on WU and P' through 
the other two; and if H, O' denote the focal lines of the four cones of recent articles, 
we see that the tangent planes to the quartic cone Q of art. 126, which touch it at the 
circular points at infinity, intersect two by two in the lines FI, IT'. Hence we have the 
following theorem: — The focal lines of the four confocal cones of Q are the double focal 
lines of Q itself. 
Cor. 1. Since every quadric has six planes of circular section, including real and 
imaginary, ice infer that the cone Q has six double focal lines. 
Cor. 2. Since the points in which the focal lines of Q intersect the sphere U are double 
foci of WU, it follows that every sphero-quartic has six double foci. 
131. The theorem of the last article may be established as follows. Any plane I will 
cut the sphero-quartic in four points ; these points are common to the four cones pass- 
ing through the sphero-quartic. Hence, by reciprocation with respect to U, through 
any point i can be drawn four planes to intersect the planes of the nodal conics of 2, 
each in four lines, forming four tetragrams described about the nodal conics. And 
since each tetragram has six angular points, from any point i can be drawn six lines 
piercing the planes of the nodal conics each in six points, which will be the angular 
points of tetragrams described about the nodal conics ; and by supposing the plane I to 
be at infinity, the point i will be the centre of U, and the six lines will be the six focal 
lines of the four confocal cones. — Q.E.D. 
132. When the circle of inversion a, touches the focal sphero-conic F, the sphero- 
quartic has a double point, and it is the spheric inversion of a sphero-conic (see art. 81),. 
or the ordinary inversion of a plane conic from a point outside the plane of the conic 
and the cyclide WU, which will be got from the equation of the sphero-quartic by 
putting spheres for circles, as previously explained, will be the inversion of a central 
quadric* Again, the quartic cone Q, got by substituting single sheets of cones for the 
circle, will have a double line and three focal cones. 
133. When «is an osculating circle of F, the sphero-quartic has a cusp. This species 
of sphero-quartic is the spheric inversion of a spherical parabola, that is, a sphero-conic 
-whose major axis is, a quadrant, or the ordinary inversion of a plane parabola from a 
point outside the plane of the parabola. The cyclide WU will be the inverse of a non- 
central quadric, and the cone Q will have a cuspidal edge, and but two confocal cones. 
134. It is shown in art. 28 of ‘ Bicircular Quartics’ that the equation of every bicir- 
cular can be written in the form 2)2'=/UC, where 2, 2' are circles whose centres are the 
double foci of the quartic ; and it is easy to see that this is equivalent to the equation in 
elliptic coordinates, 
[jj 2 —v 1 =k\/ C. 
Hence by inversion we see that any sphero-quartic can be written in the form 
%X'=k 2 C, 
( 100 ) 
