Geoff
See a really good explanation here
Fuel gauge function
Most if not all Smiths fuel gauges from the early to mid sixties were like this. The four resistances of a Wheatstone Bridge are made up of three fixed ones in the gauge and one variable one in the sender. I had to fix the one on my Healey 100 a few years back and it is just the same. I found the best way to check that the sender and gauge are OK is to remove them from the car and wire them up per the diagram in the above link. The case of the gauge and the metalwork of the sender need to be connected together and to the battery positive, assuming your car is still positive earth. The gauge terminal that has the Yellow wire goes to Battery negative. The Green wire goes between the other gauge terminal and the terminal on the sender. If everything works then you have a wiring loom problem. This is also a great way to calibrate the gauge so that empty is empty and full is full. The gauge on my Healey had a break in one of the gauge coils. I repaired it by unravelling one turn of the winding and resoldering it to the post. I then had to recalibrate it by slackening the coil fixing and moving it slightly. My 403 system wasn't working either. In this case the problem was purely mechanical - the sender was completely seized up with metal corrosion. As an aside I have also had to dismantle and rebuild the solenoid that operates the fuel reserve. This was also seized up with corrosion.
Regarding the 405 I've no idea if this is the same. Gauges on the 401 and 403 are of the type I have described. One can tell this type because the needle follows the sender behaviour exactly including when the petrol slops from side to side round corners. Later type gauges have inherent damping - I think these work on some sort of heating effect but I've not had to mess with one of these so don't quote me.
Hope that helps.
Rgds
Mike Brooks
403-1402
Scotland