DE. J. CASEY ON CYCLIDES AND SPHEEQ-QUAETICS. 
587 
4. If W=(«, b, c, d, l, 7 ii , n, p, q, 7'fa, (3, y, ci) 2 =0, where a = 0,(3 — 0, y= 0, 2> = 0 are 
the equations of four given spheres, which I shall by analogy to known systems call the 
spheres of inference, then W=0 is evidently the most general equation of a surface of 
the fourth degree, having the imaginary circle at infinity as a double line. Such a surface 
has been called by Moutard an “ anallagmatic surface,” and by Darboux a “ cyclide” 
(see ‘ Comptes Kendus,’ June 7, 1869). I shall adopt the latter name. 
5. The cyclide W =0 is by the usual theory the envelope of the sphere 
cca-\-y^-\-zy-{-idh—^, 
where x, y, z , w are variable multiples, provided the condition holds: 
a, 
n, 
m, 
1A 
T 
vl/ 1 
n, 
h. 
l. 
<L, 
l/’ 
m, 
h 
G , 
r, 
“ 5 
- 0 . 
<Z > 
r , 
d, 
w, 
x , 
^ 5 
w, 
0; 
now the sphere xu-\-yfi-{-zy-\- , w6 = 0 cuts orthogonally the Jacobian of u, (3, y, and 
the equation (7) is the equation of a quadric. Hence we have the following theorem : — 
A qucirtic cyclide is the envelope of a variable sphere whose centre moves on a given 
quadric , and which cuts a given fixed sphere orthogonally. 
6. If the equation of the cyclide be of the form 
{a, b, c,f, g, hfa, (3, yf=0, (8) 
it is shown, as in the last article, that it is the envelope of a variable sphere whose centre 
moves along a plane conic and which cuts a given fixed sphere orthogonally. Now from 
the form of equation (8) it is evident that this species of cyclide has two nodes, namely, 
the two points common to the three spheres of reference a, {3, y, and that these nodes 
are conic nodes, that is, nodes which have these points as vertices of tangent cones to 
the cyclide. I shall call this species of cyclide a binodal cyclide *. 
Section II. — Generalization of methods of Section I. 
7. The results of Section I. admit of important generalization, to the exposition of 
which I shall devote a few articles. 
Let S*— A = 0, S J — B = 0 be two quadrics inscribed in the same quadric, 
S=r ! + t y 2 + z 2 + w 1 , A and B being the planes ax + a'y -f- a!’z -f- a’"w = 0 and 
bx-\-b'y-\-b"z-\-b"'w=Q respectively; we see that S J — A+#(S*— B) is the equation of 
a quadric inscribed in S, and passing through one of the conics of intersection of S — A 2 
and S — B 2 , namely, through the common intersection of these two quadrics with the 
plane A — B = 0. But if we clear (S* — A)+J’(S= — B) = 0 from radicals, the discriminant 
* [The cyclide (S) must, from the form of the equation, have two nodes ; hut in certain special cases which 
will be discussed in the sequel, it will have one or two additional nodes. — January 1872.] 
MDCCCLXXI. 4 M 
