Following my recent ignominious extraction from the Dupin node quicksands with the assistance of a flying-squad who do know some algebraic geometry, and muster a computer algebra system which can solve polynomial equations, I remounted my cyclide and pedalled off shakily to further adventures in neglected corners of this subfusc mathematical suburb. Among the customary extensive selection of technical lamposts, potholes and manhole-covers lying subsequently in wait, there lurked the following conundrum. To recap a recent post by Ralph, a Dupin cyclide in canonical pose may be specified by three parameters (due apparently to Maxwell, rather than to Cayley as earlier hazarded). The radius "a" specifies a central circle at the origin; with centre on this, a perpendicular cross-sectional circle sweeps out a tube with mean radius "m"; while in the course of a single revolution, the actual tube radius is offset from m-c through m+c and back to m-c again. In terms of c,a,m, the implicit equation is then a Cartesian quartic (x^2 + y^2 + z^2 - m^2 - a^2 + c^2)^2 - 4(a x - c m)^2 - 4(a^2 + c^2)y^2 = 0 . The general shape of the surface will be "horned" when 0 < m < c < a ; "ring" when 0 < c < m < a ; "spindle" when 0 < c < a < m ; with various special cases at boundaries between. For example, when the differential offset vanishes c = 0 we have a torus; when in addition the tube radius equals the central radius m = a we have a double sphere centred at the origin (and giving the average graphics surface plotter a furry tongue). Now consider the special case where offset equals central radius c = a. It's easily established that the equation factors as the product of (x - 2a)^2 + y^2 + z^2 - (m - 2a)^2 , with (x + 2a)^2 + y^2 + z^2 - (m + 2a)^2 ; the surface comprises two tangent spheres with centres on the x-axis (by the way, as oriented spheres properly tangent, not anti-tangent). 'Ang abaht tho' --- one definition of a Dupin cyclide is the envelope (unique when spheres are oriented) swept out by a sphere moving tangent to 3 fixed spheres. For what 3 spheres (including planes and points) could such an envelope possibly be a pair of tangent spheres? Fred Lunnon