https://www.youtube.com/watch?v=M_BUn4TDns8 demonstrates. And there is a proof, which appears momentarily in the video at 3:35, if you freeze the video and view at full screen magnification you can actually try to read the proof. The first line in the proof is the necessary condition about tooth count sum which makes it work. This is quite a beautiful thing, which is highly non-obvious, but has a very simple pre-college ancient-Greek sort of proof. However, if the radii b and b' are unequal then Somsky's proof indicates the two gears with centers B and B' will rotate at constant angular velocity hence necessarily eventually collide. Therefore, this kind of planetary gear is IMPOSSIBLE if all gears coplanar and you want to be able to rotate the central gear (center C) an infinite number of times without being killed by a collision. By making the gears have suitable nonzero thicknesses and with the gears with centers B' and B being thinner and in sufficiently offset parallel planes, then they can "pass through" each other rather than colliding... but that will not save us because gear with center C will be unable to mesh with them both. So, I conclude this kind of planetary gear is not possible unless the travel distance is restricted; you cannot do an infinite number of rotations. So it seems practically useless, unfortunately, except in the well known symmetric case where all planets same size. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)