ORIGINAL: ytell
Sorry for being slow on this...
1. If CG is calculated about empty CG, than the arm is always zero and fuel amount has no affect.
2. If CG is calculated about LE, then the moment of fuel tank is the product of the arm (constant) times the weight of the fuel (changing). As a result, the CG will change as fuel is consumed. Am I missing something?
By "calculate" the CG, I assume you mean determine where the CG is supposed to be. If so, that calculation has nothing to do with whether the tank is full or empty. When you determine whether the CG is where it's supposed to be, you normally do it with the tank empty because the tank is usually in front of where the CG is supposed to be. But if the tank is on the CG (i.e. on the point where the CG is supposed to be), it doesn't matter whether the tank is full or empty when you check to make sure the CG is where you wanted it.
So there is no contradiction. If the tank is on the CG, no change in CG when it empties.