DB. J. CASEY ON CYCLLDES AND SPHERO-QITAKTICS. 
691 
of WU ; this pair of points will be inverse to each other with respect to the vertex of 
the cone. Hence a point on the conic corresponds to a pair of inverse points on the 
sphero-qnartic. Again, a point and its polar with respect to the conic correspond to 
a pair of inverse points and a circle, which are related to each other with respect to the 
sphero-quartic, as poles and polars are in ordinary geometry. For example, the point 
and polar with respect to the conic are such that any line through the point meets the 
conic in two points such that tangents to the conic drawn through them meet on the 
polar; and the pair of inverse points and their polar circle are such that any circle 
through the inverse points meets the sphero-quartic in two pairs of points, such that 
the generating circles which touch the sphero-quartic at these points intersect on the polar 
circle of the pair of inverse points. 
If we have any system of collinear points on the plane of (b — a)y 2 -\-(c — a)z 2 -\-(d— a)w 2 , 
it is evident we shall have to correspond with them a system of inverse pairs of points 
which are concyclic. Lastly, to a system of concurrent lines we shall have a corresponding 
system of coaxal circles on the sphere U. 
272. From the last article we see that, being given any graphic property of the conic 
( b — a)y 2 + (c — a)z 2 + (d — a)vf = 0 , 
we shall get a corresponding graphic property of the cyclide WU by the following 
system of substitutions : — 
(b-a)y' 2 -\-(c— a)z 2 a)w 2 . 
(WU). 
I A 
a. For a point on, 
b. A point having any permanent 
relation to, 
c. A system of collinear points, 
{ n! 
II. i 
A. A pair of inverse points. 
B. A pair of points having a corresponding- 
relation. 
C. A system of concyclic inverse pairs of 
points. 
A'. A generating circle of 
B'. A circle having a corresponding rela- 
tion to 
C 1 . A system of coaxal circles. 
273. If we take the reciprocal of the conic (b — a)y 2 -\-(c—a)z 2 -\-(d—a)w 2 =0, that is, 
the conic in tangential coordinates (b — — a)v 2 -\-(d — «)§ 2 , we get properties of 
WU, by substitutions, reciprocal to the foregoing ; hence we are to substitute from the 
last article, for 
a, b, c; A', B', C'. 
a', b', d ; A, B, C. 
a'. A tangent to, 
V . A line having any permanent 
relation to, 
\d. A system of concurrent lines. 
Cor. 1. Hence, being given any graphic property of a plane conic, we can get two 
correlative properties of a sphero-quartic. 
Cor. 2. The properties of bicircular quartics which are derived by substitutions from 
those of conics have their analogues in sphero-quartics. 
Cor. 3. If two sphero-quartics have one centre of inversion common to both, they 
MDCCCLXXI. 5 B 
