622 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
clearly we must have 
given by (306) 
is a 
a, 
h, 
9’ 
l> 
a. 
h, 
b, 
m, 
y, 
f 
c, 
n, 
7 
i, 
m, 
n, 
d, 
3 
a, 
A 
8, 
0 
= 0. 
(307) 
Hence the locus of the centres of all bitangent spheres of the system orthogonal to 
^=0 is the quadric surface given by (307). This surface is called by Casey the 
“ focal quadric.” We see that the focal curve on the principal sphere ^=0 is the 
curve of intersection of the sphere with the focal quadric. 
282. The surface (305) is a cubic cyclide, if it is satislied by the coordinates 
(l, 1, 1, 1) of the plane at infinity; it follows that when this is the case the quadric 
(307) is touched by the plane at infinity, and, therefore, the focal quadrics of cubic 
cyclides are paraboloids. 
283. If we take for our spheres of reference the other four principal spheres, the 
equation of the cyclide takes the form 
ax 1 + by'~ -f cz 2 -f- div 2 = 0, 
and then the focal quadric is given by 
?:: + ^ + r + * i =u . 
a'b' cw 
so that the tetrahedron formed by the centres of any four principal spheres of a 
cyclide is self-conjugate with respect to the focal quadric corresponding to the fifth. 
284. In the case of a Cartesian, the focal quadrics must be spheres : hence, if 
A, B, C, D be the centres of four bitangent spheres which we will take for our spheres 
of reference, then since the focal quadric must be of the form 
Cl 2 j3y + b' 2 y a c : u./3 a'~aS -j- b'~/3S -f- c' 2/ yS = 0, 
where a, b, c, a, b', c are the sides of the tetrahedron ABCD, the equation of the 
cyclide will be 
0, c 2 , ¥, 
O r\ O 
<r. 0, or, 
b\ « 2 , 0, 
a'\ U\ c'\ 
j., y, z , 
a' 2 , x 
y 
to 
C 3 Z 
0, w 
W, 
= 0 : 
(308) 
0 
