MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
615 
And this plane v=0 must clearly pass through the centres of the other spheres. 
Suppose now this curve to represent a circular cubic, then 
— + 1 1 + - — ~ ] ~ 
r-f ct — e'r^ 5 — c'r.^ c — e'r / d~e 
(287) 
The curve then passes through the centres of the principal spheres ; and since these 
points will then satisfy the condition of being foci, the imaginary circle at infinity 
must be a cuspidal edge on the cyelide. A cyclide of this nature is called a Cartesian. 
264. If the coefficients of the equation 
ax 3 dur+e v 2 = 0, 
are connected by the relation 
the surface represents a cubic cyclide. 
Regarding this as the general cubic cyclide, we see that it passes through the 
centres of its five principal spheres. 
The equation of the tangent plane at the centre of the sphere £c=0, being 
(a-b) ( a — c ) {ci — cl) (a — c) 
yd- —— -z- b —— -w-\- v=0 ; 
(288) 
which is clearly parallel to the plane 
ax by cz dw ev 
-+r+r + -r+r=0- 
Thus the tangent planes to the cyclide at the centres of its five principal spheres 
are all parallel, and they may be regarded as the five tangent planes to the cubic from 
the line at infinity on the surface. 
(B.) General Nodal Cyclide .—§§ 265-271. 
265. We have seen in § 253, that the equation of a cyclide having one node can be 
expressed in the form 
ax 3 + hy 2 +c 2 3 +dz«; 3 = 0. 
The system of reference being three orthogonal spheres ( x , y, z) and their two points 
of intersection (w, v): so that the equation of the absolute is 
x 3 +y 3 +z 3 = iwv ; 
