No doubt about that! So, is the Michelson-Gale-experiment you refer to this one (from wikipedia):
The Michelson-Gale-Pearson experiment is a modified version of the Michelson-Morley experiment which tests the aether drag along the rotating frame of Earth. That is, if aether is dragged rotationally by the Earth, as was assumed by many, the Michelson-Gale-Pearson experiment would be able to detect this effect.
[snip]
Interestingly the experiment was expected to generate a positive result both for an entrained aether as well as due to relativistic effects. The main difference would be the magnitude of the effect. It was thus a surprise to everyone when the MGP experiment also returned what appeared to be a null result, or at least a rather inconclusive one. The results consisted of 269 measurements that showed an effect of -0.04 to +0.55 fringes, which could be seen as evidence of the rotational effects, but at the same time they were not statistically significant.
BTW - what's about the moon?
No that's not it.