difference

Shown is the tall concentric meter. We dont have a complete clock for the early B/M water meter.

In general U bend of mercury, the differential pressures push either side of the mercury and floats are used either side, like the wheel barometer, to give an output to a meter.
Sounds simple, its not, they are huge great chambers that have to be sealed, to cope with the high pressures of the water, or later systems the isolation fluid.
Shown is the differential amplifier sometimes called the diff. amp., u bend and various other things.

Points to note are;

1) The drain at the base to draw the mercury
2) The equalisation tap in the centre.
3) The two output pulley at the lip of the cylinders. These are couples to racks which drive the instruments above.

The input is the fluid and this is the differential input. The output is the rotation of the wheels, which are counter balanced a bit like a sash window is. So float movement can be used.

Well, we now have a mechanical output, in this case the pulleys that rotate and drive through a rack to the meters at the top. This now leads us to the mechanical calculator, the need for a time base, clock, a way of recording the pressure and the differential pressure and how are we going to get to volumetric measurement. Pressure proportional to flow, and flow /second = volume. Time for the pencil and paper and have a guess.