686 
DE. J. CASEY OX CYCLIDES AND SPHEKO-QUAKTICS. 
correspond a hinodal cyclide circumscribed about W; the nodes of the binodal cyclide will 
correspond to the plane of the conic. 
262, Since to any system of planes passing through a point corresponds a system of 
homospheric pairs of inverse points, to a cone circumscribed about the focal quadric of a 
cyclide W will correspond a sphero-quartic on W ; and if the cone be one of revolution , 
the sphero-quartic will become a spliero-Cartesian. 
Cor. 1 . If the vertex of the cone beat infinity , that is, if the cone become a cylinder, to 
it will correspond a bicircular quartic ; and if the cylinder be one of revolution, to it will 
correspond a section of the cyclide, which will be a Cartesian oval. 
Cor. 2. Since two cylinders of revolution can be described about a quadric, through each 
centre of inversion of a cyclide can be drawn two planes which will intersect it in Cartesian 
ovals. 
263. Since to a point on F corresponds a generating sphere of W, to the line joining 
two points on F will correspond the circle of intersection of two generating spheres ; and 
if every point of the line be on F, every point of the circle will be on W. Hence to a 
rectilinear generator of F will correspond a circular generator of W ; and since through 
any point on F can be drawn two rectilinear generators, hence in general can be drawn 
two circular generators corresponding to each focal quadric of W through any point of W. 
264. The last article maybe established differently as follows. Thus if perpendiculars 
from the centres of the spheres of reference a, [3, 7, of the cyclide W=«a 2 -\-b(3 2 -\-cy 2 -{-db' i 
on any plane be denoted by X, [m, v, g, then the points whose equations are 
A, B, C, D, 
A', B', O', D', 
(L g) 
correspond to the spheres whose equations are 
A, B, C, D, 
A', B', C', D', 
(a, (3, 7, &), 
and consequently the line joining the points corresponds to the circle of intersection of 
the spheres. Now the six determinants of the matrix 
A, 
B, 
c, 
D, 
A', 
B', 
O', 
D', 
or their mutual ratios, are called by Professor Cayley the six coordinates of a line in 
space, and are denoted by the notation (a, b, c,f, g, h). Hence we see that we can in 
our extension call the same ratios the six coordinates of a circle in space (see Cayley 
“ On the Six Coordinates of a Line,” Cambridge Philosophical Transactions, vol. xi. pt. 2). 
Hence the same investigation which in Professor Cayley’s system proves any property 
of a system of lines in space, will, with our interpretation, give a corresponding property 
of a system of circles in space. It is to be remembered, however, that all our circles 
are cut orthogonally by the same sphere, namely the sphere U, the Jacobian of a, [3, 7, S 
