Inside the mesh

A rack tooth has straight sides, sloping 1 in 4 from the vertical. The pinion teeth are curved: their sides are involutes, the curve traced by the end of a string unwound from a drum. An involute rolling on a straight flank keeps the push in one fixed direction however deep the teeth engage, so the mesh stays correct as wheels and rails wear. Riggenbach adopted this tooth form on the advice of the Zürich professor Carl Culmann, and every vertical rack since has kept its flank angle.

That fixed direction is the line of action. It is tilted, so every push has two parts: one along the track that drives the train, and one upward that tries to lift the pinion out of the rack.

With one bar, a single pair of teeth carries the whole load for part of every pitch, and the load jumps from tooth to tooth as they hand over. Carl Roman Abt's answer was to lay two (or three) thin bars side by side, each staggered by a fraction of a tooth, with a pinion disc on each. There is then always a tooth in full engagement somewhere, the hand-overs are spread out and the drive runs smoothly. With three bars staggered by a third, and two pinions on the engine, a fresh tooth comes into mesh every 20 mm.

There was a catch. With the teeth staggered it is hard to make every pinion disc bear at once, so Abt's discs are mounted on springs that let each one turn a little against the others until all share the load.

A quarter of every driving push tries to lift the pinion out of the rack.

The lift is the danger. If the rack sags between its supports, or the track settles, or a train comes down too fast, the pinion can ride up and out of the teeth. Chapter 8 tells what happened on Snowdon's opening day.