= Reverted gear train
A <reverted gear train> returns the output rotation axis to the input axis, usually through a two-stage <compound gear train>. The input and output gears can rotate independently on coaxial shafts, while the intermediate gears share another rotating body. Coaxial axes are a layout condition in addition to the desired <gear ratio>.
For standard unshifted <spur gears>, let gears1 and2 mesh in the first stage and gears3 and4 in the second, with2 and3 rigidly connected. The common intermediate-axis separation requires
$$
\frac{m_{12}}2(N_1+N_2)=\frac{m_{34}}2(N_3+N_4).
$$
If both stages use the same <gear module>, this reduces to $N_1+N_2=N_3+N_4$. The reduction ratio independently satisfies $i=(N_2/N_1)(N_4/N_3)$. Both conditions must be satisfied; multiplying arbitrary stage ratios does not ensure a reverted layout.<KHK Gear Nomenclature>[Basic Guide B §1.1, p.5, compound-stage ratio; §2, p.17, standard center-distance equation(2.11)]{ref}
For an original design example, choose counts20,60,16 and64, with module $2\,\mathrm{mm}$ in both stages. Both sums are80, so both separations are $80\,\mathrm{mm}$. The reduction is $3\times4=12$, taking an input of1440 revolutions per minute to an output of120. Two external meshes give the same input/output rotation direction.
The gears occupy different axial planes so that coaxial input and output do not require them to be one rigidly joined gear. A practical design still needs independent shaft support, compatible tooth geometry in each stage and adequate clearances. Operating with shifted tooth geometry or another mounting changes the applicable separation equations; the simple tooth-sum condition belongs to the stated standard-gear model.
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