 Simple gear train

A simple gear train transmits rotation through individual intermediate gears, each intermediate gear engaging its neighbors without a separate, differently sized gear rigidly sharing that intermediate rotation. For external meshes on fixed parallel axes, consecutive signed ratios multiply and intermediate tooth counts cancel:where $m$ is the number of external meshes. An intermediate idler gear changes placement and potentially output direction, but not this ideal endpoint tooth-count ratio.^[ref]

For example, a chain with tooth counts 18, 30, 42 and 72 has three meshes andAt input speed $800$ revolutions per minute, output speed is $200$ revolutions per minute in the opposite direction. Removing one intermediate gear while maintaining a compatible two-mesh layout preserves the fourfold reduction but reverses the direction conclusion.

The cancellation concerns kinematics, not every engineering consequence. Intermediate tooth counts affect physical sizes, axis positions and angular speeds of intermediate rotating bodies; additional meshes and supports can change gear efficiency and backlash. Moreover, if two different intermediate gears are rigidly coupled, their tooth counts do not cancel by the same argument: that arrangement is a compound gear train.