MR. R LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
G17 
268. From the form of the equations of the focal curves we infer that all surfaces 
represented by the equation 
y 
vr 
ot? + k /3 2 + k 7 2 + k §" 
= 0. 
are confocal with the surface 
c) o o o 
* _iX+-+-=o • 
o 1 no HP o cvo 5 
cc p" y~ o z 
and we infer that given any cyclide having one cmc-node, through any point three 
other cyclides can be drawn having the same node and the same focal curves : and 
these cyclides cut orthogonally. 
We also see, by choosing h so that 
1 1 , 1 1 | 1 1 | 1 1 
« 2 + k r* + /3 2 + k r 2 2 + 7 2 + k r 3 2 + S 2 e 2 _ ‘ ’ 
that three cubic cyclides can be drawn having the same node and focal curves as a 
given nodal quartic cyclide. 
269. If we have b=c, we have seen that the equation 
ax~ + by 2 + bz z + dw 2 = 0 
represents a trinodal cyclide, w— 0 being one node : its equation may also be written 
or 
(a -p b )x 2 -}- d iv 1 +4 bwv = 0, 
aV+2/iro= 0, 
where u is a sphere passing through the other nodes, but in this case the equation of 
the absolute will not be so simple. Taking the first form of the equation we see that 
the system of bitangent spheres is given by 
£=0; 
ri + f 2 , 4wnr d 0 _ A 
(293) 
and the corresponding focal curve by 
A d A 
, t = 0 ; ■■ — tv=0, 
b—a a 
(294) 
which is a circle on the principal sphere :r=0. 
270. If one of the principal spheres, say 2 , become a plane, the corresponding focal 
curve is a plane nodal bicircular quartic, and if it pass through the centres of the other 
two principal spheres, we have 
a b_ J-1 
a — c rd b — cr 2 2 c 2 e 2 
(295) 
MDCCCLXXXVI. 
4 R 
