I asked earlier whether it is possible that the envelope of a sphere moving tangent to to three fixed spheres can comprise two tangent spheres. The question is manifestly unfair: to begin with, how many people can visualise the envelope of a sphere moving tangent to three fixed spheres, anyway? Which exposes one major weakness of this oft-quoted synthetic definition of a Dupin cyclide --- elegant, but unusable. At the cost of a certain amount of algebraic machinery an alternative approach provides more insight, including a practical construction counterpointing this tangency constraint. The prerequisite is essentially "hexaspheric" coordinates, as adopted by XIX-th century geometers such as Coolidge for "oriented spheres" or "contact" geometry --- a modern reference is T. E. Cecil "Lie-sphere Geometry" (2008). The details are not relevant here: what matters is that it provides a framework representing extrinsically oriented 3-space points, spheres, planes, utilising coordinate 6-vectors. Now in classical line geometry, a 3-space quadric hyperboloid may be regarded in two ways as a linear system generated by 3 skew lines lying on the surface, lines being represented by Pluecker 6-vectors. Analogously, a Dupin cyclide is simply the linear regulus generated by any 3 distinct spheres along its envelope. Furthermore, a given cyclide is generated by two distinct systems: if one system is thought of as a path swept by a continuously moving sphere, the other comprises all spheres to which the motion remains tangent; and vice-versa. The isometries of contact geometry combine Euclidean (rigid), Moebius (conformal) and Laguerre (equilong) transformations; elementary continuous motions generalising rotation include dilation, offset, and many more intriguing things ... but I digress. A Dupin cyclide may be regarded as the envelope of a sphere under the action of one of these elementary rotors; and now at last we can get to the point. More precisely, to the point of tangency of two spheres proposed as a special case of a cyclide, when swing offset c equals central radius a. Under rotovation (as it were) subsequent motion is determined completely by current location, so each position along the orbit is visited (at most) once during the period; whereas a sphere sweeping the two-sphere envelope must shrink down twice over to pass through the tangency point. We conclude that from a synthetic aspect the class of cyclides is not compact, and is properly contained within the class defined analytically via their implicit Cartesian equation. In particular, a tangent sphere pair is analytically a cyclide, but synthetically only a limit of a cyclides. Incidentally the same objection can also be raised in connection with the double sphere with two double points, mentioned earlier as a special case of a torus when tube radius m equals central radius a. Although this case is easier to visualise, it seems harder to equip with a convincingly rigorous argument. Fred Lunnon