614 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
261. If cl=e, then the equation of the cyclide may be written 
ax 2 -}- by 2 -}- cz 2 = 0. 
It has clearly two cmc-nodes : the points common to (x, y, z ) ; and we have seen in 
§ 254 that one of these spheres is imaginary. Also, the surface has three distinct 
principal spheres ; but any pair of orthogonal spheres cutting (x, y, z ) orthogonally 
may clearly be considered as principal spheres. The case corresponds to that of a 
quadric of revolution. 
The coordinates of a bitangent sphere at the point ( x'y'z'w'v ) must satisfy the 
equations 
£ TJ £ CO CT 
(a + k)x' (b + k)y' (c + k)zl lew' lev r 
There are only three systems of bitangent spheres, viz.:— 
O OO 0 , o 
f=0. ^-+-A+“-^ =0 
a —b a—c a 
s-o 2 ■ o 
a r , ? , o)-+®- A i 
r) —— 0, +-p—;— = 0 f 
' b — ci b — c b 
5-2 2 2 i 2 
1=0, L.+jl-+2Jz .=o 
c—ct c—b c 
(283) 
The focal curves on the principal spheres will be circles; they are given by 
x= 0, - - y 2 -}—-- z 2 = 0 
a — b J a—c 
y= o, 7 -^-x 2 +w- 2 2 =o 
,y o—a o—c 
2=0, -ar+— -y ~ j =0 
c—a c—b J 
(284) 
262. Again, if 7; = c, oZ=e, we have seen that the cyclide has four cm'c-nodes, and 
one principal sphere x, which is imaginary. There is one system of bitangent spheres, 
given by 
£=o, 
Q , Q , 2 
V + b‘ . to 
a — b a —cl 
(285) 
263. Let us suppose now the radius of one of the principal spheres to be infinite, 
say r 5 ; the corresponding focal curve is a plane bicircular quartic, its equation being 
w- 
d — e 
= 0 
(286) 
