I found this A²h ( Ampèrequadraaturen )meter while doing a clean up in my basement.
I got it from a power utility in The Netherlands in the 80's to add to my collection.
Any ideas where this was used for?
On DC applications Ah meters are used but this is an AC meter.

What is the reason for measuring Amp x Amp x Hour?

Amp x Hour x Voltage equals energy (Power x Time).

What does Energy x Current equal?

Larry, I think Energy x Current probably belongs in another dimension! "Set phasers to exterminate!"

Electric power = current times the potential difference;
Unit of power: watt = (amperes x volts)

P = IxV : Power = current x voltage.

but,

P = IxI x R ; Power [loss] = current squared x circuit resistance.

Peering into my Dilithium Crystal Ball, could this meter have been used in an electric traction operation? Measuring Amps squared x hrs on a known resistance of track/overhead wires [got from the voltage drop] could indicate the non-traction losses with no traffic = leakage/signalling etc., over a timebase. Compared to circuit losses with traffic, giving an efficiency factor. The actual resistance of the track would be treated as a constant. Could the same meter be used on other power transmission circuits, within voltage limits, for power loss assessment? That might explain why it only has a simple additive totals scale.

http://www.ehow.com/facts_6855370_types-quadrature-oscillator.html

This device is specifically designed for broadcast circuits and probably was used to assure the authorities that the emitter was not exceeding permitted power levels.

Ampere-Quadrature Meter being a pretty straight translation.

The A^2 refers to the phase displacement.

Interesting hmmm, It crossed my mind that it was perhaps used on the Dutch railways, although they run on 1500 Volts DC.
The AC side would normally been metered with kVA metering instead of amp sqrd hours.

Tesla, the transmitter thought is very interesting.

Unfortunately the power authority I got this meter from doesn't exist anymore, and all staff I used to know have retired by now.