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Dear Daves and Greg, thank you for your comments and advices!

Last night a made an other trial, although the circumstances were far from ideal (some haze and fast moving clouds) I was able to make a new series. Jupiter was out of my view, so Altair was selected as westerly body (actually that was the only star that I could figure out near the ecliptic), Mars as easterly. I could not perform the same accuracy, but still (let's allow) acceptable individual measurements. And the average of them is again much better (15 seconds of time difference).

Greg, I took your advice, and made some analysis on the effect of the variation of index error. I do guess that 20 moa variation is too much for lunars, we talk here about fraction of moas, so I made my check with finer resolution. The effect on the individual measurements is significant (more than 10 seconds of time per 0,1' change in index error) but has definitely opposite signs for the east and west measurements in the range of -2.0' +2.0' of index error variation. And thus the average of them changes by a fraction only in this range.

However the variaton of the average follows the variation of the individual measuremenst for larger variations of IE (10 moa).

I attach my analysis: there are two sheets for each nights, and at the end of each sheet a data table to demonstrate the effect of index error variation. The problem is beyond my mathematical knowledge, I just have some guesses here. It might be that I used Young's method that is valid for spherical Earth, and this left-right attempt cancels out some distortion caused by the non-spherical Earth? I will keep evaluating this.