690 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
through r points, to touch s lines, and to touch t planes, where r+s + t— 9, then pre- 
cisely the same number N of cyclides can be described, being given r generating spheres to 
have double contact with s circles and to pass through t inverse pairs of points. 
269. If Y be the reciprocal of the focal quadric E with respect to U, or, in other 
words, if V be a quadric of the system passing through the spliero-quartic WU, then 
the planes, lines, and points of V will correspond to the points, lines, and planes of F ; 
and hence by substitutions reciprocal to those of art. 265, being given any graphic pro- 
perty of V, we can get a corresponding graphic property of W. 
Cor. Hence, being given any graphic of a quadric, we can get two correlative graphic 
properties of a cyclide. 
Section II. — Substitutions. Sphero-quartics. 
270. We have seen, if in the equation of a cyclide W =<£>(«, (3, y, b), where (p represents 
a homogeneous function of the second degree, we regard a, /3, y, ^ as the small circles 
in which the spheres «, (3, y, intersect U, that we get the equation of the sphero- 
quartic (WU) ; also that the spliero-quartic is generated as the envelope of a variable 
circle, whose centre moves along a sphero-conic, and which cuts a given circle ortho- 
gonally ; and we might investigate, as in the last section, a system of substitutions by 
which, from known properties of sphero-conics, we could infer properties of sphero- 
quartics ; but there is a simpler system of substitutions by which we may arrive at the 
latter, namely, by means of substitutions from known properties of plane conics. This 
method I shall explain briefly in the following articles. 
271. Let W=ffa 2 + £/3 2 + cy 2 + (7c> 3 =:0 be a cyclide, then the Jacobian of a, (3, y, b is 
given by the equation 
U 2 ^« 2 + /3 2 +y 2 +S 2 = 0. 
Hence the spliero-quartic (WU) will be the curve of intersection of U, and either of the 
binodal cyclides 
W— «U 2 , W -HP, W — cU 2 , W — <7U 2 . 
Now let us consider W — «U 2 , or (b — a)j3 2 +(c — a)y 1 -\-{d~ a)b 2 ; this cyclide has four 
focal quadrics, of which one reduces to a plane conic, and this conic is a focal conic of 
each of three remaining focal quadrics (see art. 113). The conic is one of the nodal 
lines of the developable S (see Chapter VIII.), and is the reciprocal of one of the four 
cones through (WU). Now any tangent plane to the cone will intersect U in a circle, 
which will be a generating circle of WU, and this tangent plane will intersect the plane 
of the nodal conic of "S, that is, the plane of the conic whose equation in tangential 
coordinates is 
(b— a)yj‘ 2j t-(c—a) 2 v-{-(d—a)f, 
in a line, which wall be a tangent line to the reciprocal conic, that is, to the conic whose 
trilinear equation is (b—a)y 2 -\-(c—a)z 2 + (d—a)iv 2 = 0. Hence a tangent line to this 
conic corresponds to a generating circle of WU. Again, any edge of the cone intersects 
the conic (b — a)y 2 -\-(c— a)z 2 -\-(d— a)w 2 in a point, and passes through a pair of points 
