76 
ON THE CYCLIDE. 
[569 
It may be remarked, that if we endeavour to generalize a former generation of 
the cyclide, and consider the envelope of a variable sphere having its centre on a conic, 
and touching a fixed sphere, this is in general a surface of an order exceeding 4; 
it becomes a surface of the fourth order, viz. a cyclide, only in the case where the 
fixed sphere has its centre on the anti-conic. But if we consider the envelope of a 
variable sphere having its centre on a conic and cutting at right angles a fixed 
sphere, this is always a quartic surface having the circle at infinity for a double line; 
the surface has moreover two nodes, viz. these are the anti-points of the circle which 
is the intersection of the sphere by the plane of the conic. If the sphere touches 
the conic, then there is at the point of contact a third node; and similarly, if it has 
double contact with the conic, then there is at each point of contact a node; viz. in 
this case the surface has four nodes, and it is in fact a cyclide. 
There is no difficulty in the analytical proof: consider the envelope of a variable 
X 2 F 2 
P + « 
angles the sphere (x — If + (y — mf + (z — nf = k 2 . 
1, and which cuts at right 
sphere having its centre on the conic Z = 0 
Take the equation of the variable sphere to be 
o - X) 2 + (y - Yf + z 2 = c 2 , 
then the orthotomic condition is 
(X — If + (F— mf + n 2 = c 2 + k 2 , 
or, substituting this value of c 2 , the equation of the variable sphere is 
(x — Xf + (y — Yf + z 2 = — k 2 + (X — If + (F — mf + n 2 , 
all which spheres pass through the points 
that is, 
x — l, y = m, z = ± \J{n 2 — Jc 2 ); 
x 2 + y 2 + z 2 + k 2 — l 2 — m 2 — n 2 — 2 (x — l) X — 2 (y — m) Y = 0, 
and considering F, F as variable parameters connected by the equation 
the equation of the envelope is 
(x 2 + y 2 + z 2 4- k 2 — l 2 — m 2 — n 2 f + 4$ {x — If — 4a (y — mf = 0, 
viz. this is a bicircular quartic, having the two nodes x = l, y = m, z = ± *J(n 2 — k 2 ); these 
are the anti-points of the circle (x — If + (y — mf = k 2 — n 2 , which is the intersection of 
the sphere (x — If + (y — mf + (z — nf = k 2 by the plane of the conic. 
The constants might be particularised so that the equation should represent a 
cyclide; but I treat the question in a somewhat different manner, b} 7 showing that 
the generating spheres of a cyclide cut at right angles each of a series of fixed 
spheres. Write a, /3, y = h 2 — c 2 , c 2 — a 2 , a 2 — b 2 ; then if 
F 2 X 1 2 _^ 2 _ 
a ’ y a 
the points (X, F, 0) and (X 1} 0, Zf) will be situate on a pair of anti-conics.