I'm not sure I made my ...

I'm not sure I made my problem clearly enough.

When using a meter to bill for consumed water it is important that the meter registered value is the true consumption. The consumer will be billed at a unit cost based not simply on the water consumed but also for all the associated costs.

The inferential meters, first used for direct plumbing, satisfied this requirement because, with a 1/4 turn tap on the incoming mains, the native flow profile was of bursts of flow at maximum flow. There is no significant flow below the meter's starting flow rate.

The positive displacement meters were necessary for indirect plumbing where a gravity tank is maintained full by a float valve. This means that while a lot of flow takes place at high flow rates, a significant proportion of flow takes place at low flows diminishing to zero. The inferential meters cannot handle this but the reciprocating piston meter (Thomas Kennedy, 1824) does. Not because it measures to zero flow but because, when the meter stops, flow stops and there is no significant unregistered consumption.

The volumetric meters (Rotary piston introduced in the 1860s) and nutating disc (Introduced in the 1880s) are only semi positive. This means that when they reach their starting flow and registration stops, the open flow paths allows some flow to continue unregistered.

This is not therefore about the accuracy of measurement within the operating flow range of the meters but about what happens below the starting flows. A volumetric meter which registers 100m3 (+/-allowed accuracy) may actually represent a true consumption of 105M3. Equability simply means that it is the same relationship for all meters and thus the registered value can be used a a proxy for for the actual consumption.

Before this concept as introduced the volumetric meters were intended to extend the measurement capability in direct plumbing in competition with inferential meters (and perhaps why inferential meters are not used in direct plumbing systems in the principles countries of manufacture (UK and USA).

The volumetric meters have not significantly improved since the 1880s (the last of the Lewis Nash patented modifications) nor made any commercial advances, not till Kent introduced net shape manufacturing in the 1960s.

While the simplicty and low cost of the volumetric meters was undoubtedly attractive, irresistibly so, PD meters seem to have carried on into the 20th century.

My point is that the unchanged volumetric meters were unable to displace t