684 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
254. The following properties are the inverses of properties of conics &c. : — 
1°. A circular cubic is the locus of one set of foci of all the conics that can be drawn 
through four concyclic points. Hence, by inversion, a sphero-quartic is the locus of the 
locus of one set of foci of all the spliero-quartics with a double point which can he dream 
through four concyclic points. 
A more general proposition than this can be easily inferred from art. 253, 6°. 
2°. If two tangents to a conic intersect at a given angle, the locus of their intersection 
is a bicircular quartic. Hence, by inversion, 
If two generating circles of a sphero-quartic with a double point { including cusps and 
conjugate points) intersect at a given angle, the locus of their intersection, if they belong 
to the system of generating circles which passes through the double point, is a sphero- 
quartic. 
Cor. If the angle of intersection be a right angle the locus will be a circle. 
3°. A cardioide can be inverted into a cissoid. Hence a cusped sphero-quartic will be 
got by inverting a cusped sphero-Cartesian. 
255. Particular spherical sections of a general cy elide will be sphero-Cartesians. 
The following is an example: — Let W be a cyclide, U and F 'a sphere of inversion 
and corresponding focal quadric; then if any sphere has its centre on the focal hyperbola 
of F and cuts U orthogonally, it will intersect W in a sphero-Cartesian. 
CHAPTER XIII. 
Section I. — Substitutions. 
256. If W [a, b, c, d, l, m, n, p , q, Fjjjoq /3, y, c)) 2 =0 be the equation of a cyclide, 
and if the equation W be satisfied by the values x', y', z', w' of a, (3, y, h, we can state it 
thus : the system of six spheres denoted by the matrix 
u i y •> 
I, if, z', w', 
have the two points which are common to them on W ; and if the ratios ot? : f : z' : w' be 
supposed to vary, but subject to the condition 
(a, b, c, d, l, m, n,p, q, rjxf, if, z', wj= 0, 
then the pair of points denoted by the matrix (164) will vary, and the locus will be the 
cyclide W. Hence we may call [a, b, c, ... . rfjx, [3, y, §) 2 =0 the local equation of a 
cyclide. 
I remark that whenever I shall speak of a pair of inverse points on a cyclide it will 
be a pair determined by a matrix such as (164). 
257. We have seen that the tangential equation of the focal quadric of a cyclide is 
the same in form as the local equation (see last article) of the cyclide, and that to a tangent 
plane of the quadric will correspond a pair of inverse points of the cyclide, and generally 
