With planets placed in cyclic order around the sun, and axis indicated by semicolon, the vector X = [ r, s; p, ..., q; u, ..., v ] of gear radii becomes essentially unique; and combined with sunset (offset) h , it specifies the train. While in a realisable train the radii (more properly, tooth counts) are positive integers, it becomes convenient to generalise them to be rationals. In particular, my sun radius s is normally negative! Vectors X, Y of different trains are related by some sequence of concentric (coaxial) increments and rescalings just when a X + b I = c Y + d I , where I = [1, 1; 1, ..., 1] with a,b,c,d integers. X can be reduced to a unique canonical form via Z = ( X - (r+s)/2 I ) / (r-s)/2 = [ 1, -1; p', ... ] ; to establish whether trains X, Y are related, we need only compare their reduced forms! Example: the smallest 8-planet train is X = [ 28, -12; 12, 8, 4, 3; 4, 8, 12, 13 ] , h = 10 . Scaling this up by c = 2 then incrementing by d = -3 yields the train reported earlier by Somsky, Y = [ 53, -27; 21, 13, 5, 3; 5, 13, 21, 23 ] , h = 20 . But if unaware of their relation, we could deduce it by reduction of both to Z = [ 1, -1; 1/5, 0, -1/5 -1/4; -1/5, 0, 1/5, 1/4 ] , h = 1/2 . Note that for this family, largest and smallest planets lie on the axis; also we can show that for any related vector representing realisable gears, it is impossible to prevent overlap between the largest three. Reduced forms have some pleasant properties: primarily, they save space in a train catalogue. Also a "Somsky pair" of planet radii p, v , on opposite sides with r + s = p + v , reduce to radii with equal size and opposite signs; if furthermore p = v then both reduce to zero. Although reduction apparently fails if r = s , this case is the null family with planets and sunset vanishing: we set Z = [0, 0; ..., 0] . Finally (as demonstrated elsewhere in inadvertant but convincing fashion) train coordinates as defined above labour under a dichotomy between radii (sic) X and centre h : the resulting nuisance involves not merely the scale factor 2pi , but varying behaviour under symmetry transformations. It is possible to eliminate this conceptual untidiness. One strategy ditches the centres, instead incorporating belt lengths around adjacent pairs of planets into the vector. Another uses Lie-sphere coordinates to unify radius and centre: rescaling and concyclic increment correspond to geometric transformations of an oriented circle, respectively dilation and (wait for it) offset (noooh!). Fred Lunnon